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泊松分布

2021-01-11T11:59:50Z

Fairywang：

此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。
审校：fairywang
{{Probability distribution

''' 概率分布Probability distribution'''

| name = Poisson Distribution

''' 泊松分布 Poisson distribution'''

| type = mass

类型 = 质量

| pdf_image = [[File:poisson pmf.svg|325px]]

| pdf_image = 325px

| pdf_caption = The horizontal axis is the index ''k'', the number of occurrences. ''λ'' is the expected rate of occurrences. The vertical axis is the probability of ''k'' occurrences given ''λ''. The function is defined only at integer values of ''k''; the connecting lines are only guides for the eye.

| pdf_caption = The horizontal axis is the index k, the number of occurrences. λ is the expected rate of occurrences. The vertical axis is the probability of k occurrences given λ. The function is defined only at integer values of k; the connecting lines are only guides for the eye.

| pdf _ caption = 横轴是索引 k，表示出现的次数。是预期发生率。垂直轴是给定的 k 发生概率。函数只定义在 k 的整数值上，连接线指示方向。

| cdf_image = [[File:poisson cdf.svg|325px]]

| cdf_image = 325px

| cdf_caption = The horizontal axis is the index ''k'', the number of occurrences. The CDF is discontinuous at the integers of ''k'' and flat everywhere else because a variable that is Poisson distributed takes on only integer values.

| cdf_caption = The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Poisson distributed takes on only integer values.

| cdf _ caption = 水平轴是索引 k，表示出现的次数。因为一个''' {泊松分佈 Poisson distribution'''的变量只取整数值，所以 CDF 在 k 的整数和平坦的所有其他地方均不连续。

| notation = <math>\operatorname{Pois}(\lambda)</math>

| 表示法 = < math > operatorname { Pois }(lambda) </math >

| parameters = <math>\lambda\in (0, \infty) </math> (rate)

| support = <math>k \in \mathbb{N}_0</math> ([[Natural numbers]] starting from 0)

| support = < math > k in mathbb { n } _ 0 </math > (自然数从0开始)

| pdf = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math>

| pdf = < math > frac { lambda ^ k e ^ {-lambda }{ k！{/math >

| cdf = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math>

| cdf = < math > frac { Gamma (lfloor k + 1 rfloor，lambda)}{ lfloor k rfloor！{ i = 0}{ lfloor k rfloor } frac { lambda ^ i }{ i！} </math > ，或者 < math > q (lfloor k + 1 rfloor，lambda) </math >

(for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the [[upper incomplete gamma function]], <math>\lfloor k\rfloor</math> is the [[floor function]], and Q is the [[regularized gamma function]])

(for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the upper incomplete gamma function, <math>\lfloor k\rfloor</math> is the floor function, and Q is the regularized gamma function)

(对于 < math > k ge 0 </math > ，其中 < math > Gamma (x，y) </math > 是上面的不完全Γ函数，< math > lfloor k | | | | | | | | | | | | | | | | | | </math > 是下面的函数，q 是正则化的 Gamma 函数)

| mean = <math>\lambda</math>

| mean = <math>\lambda</math>

| mean = < math > > lambda </math >

| median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>

| median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>

| 中位数 = < math > > 大约1floor lambda + 1/3-0.02/lambda rfloor </math >

| mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>

| mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>

| mode = < math > lceil lambda rceil-1，lfloor lambda rfloor

| variance = <math>\lambda</math>

| variance = <math>\lambda</math>

| variance = < math > lambda </math >

| skewness = <math>\lambda^{-1/2}</math>

| skewness = <math>\lambda^{-1/2}</math>

| skewness = < math > lambda ^ {-1/2} </math >

| kurtosis = <math>\lambda^{-1}</math>

| kurtosis = <math>\lambda^{-1}</math>

| 峭度 = < math > lambda ^ {-1} </math >

| entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>

| entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>

| 熵 = < math > lambda [1-log (lambda)] + e ^ {-lambda } sum _ { k = 0} ^ infty frac { lambda ^ k log (k!)}{ k！{/math >

(for large <math>\lambda</math>)

(for large <math>\lambda</math>)

(对于大的 < math > > > lambda </math >)

<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math> <math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math>

<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math> <math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math>

12 lambda }-frac {1}{24 lambda ^ 2}-{{{} </math > < math > qfrac {19}{360 lambda ^ 3} + o left (frac {1}{1}{ lambda ^ 4}右) </math > <

| pgf = <math>\exp[\lambda(z - 1)]</math>

| pgf = <math>\exp[\lambda(z - 1)]</math>

| pgf = < math > exp [ lambda (z-1)] </math >

| mgf = <math>\exp[\lambda (e^{t} - 1)]</math>

| mgf = <math>\exp[\lambda (e^{t} - 1)]</math>

| mgf = < math > exp [ lambda (e ^ { t }-1)] </math >

| char = <math>\exp[\lambda (e^{it} - 1)]</math>

| char = <math>\exp[\lambda (e^{it} - 1)]</math>

| char = < math > exp [ lambda (e ^ { it }-1)] </math >

| fisher = <math>\frac{1}{\lambda}</math>

| fisher = <math>\frac{1}{\lambda}</math>

| fisher = < math > frac {1}{ lambda } </math >

}}

}}

}}

In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPAc-en|'|p|w|ɑː|s|ɒ|n}}; {{IPA-fr|pwasɔ̃}}), named after [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and [[Statistical independence|independently]] of the time since the last event.{{r|Haight1967}} The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

In probability theory and statistics, the Poisson distribution (; ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

在概率论和统计学中，''' 泊松分布 Poisson distribution'''是以法国数学家西莫恩·德尼·泊松 Siméon Denis Poisson命名的，是一个离散的概率分布，它表示在一个固定的时间段或空间中一定数量的事件的发生概率，这些事件以一个已知的常数平均速率发生，并且独立于与上一个事件的间隔发生时间。还可以用来表示其他有特定间隔的事件数量，如距离、面积或体积。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“它表示在一个固定的时间段或空间中一定数量的事件的发生概率”改为“它表示在一个固定的时间段或空间中，一定数量的事件发生的概率”

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution.{{r|Brooks2007}} Other examples that may follow a Poisson distribution include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution. Other examples that may follow a Poisson distribution include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.

例如，记录每天收到邮件数量的个人可能会注意到，他们平均每天收到4封信。如果收到任何邮件都并不影响未来邮件的到达时间，也就是说，如果不同来源的邮件彼此独立地到达，那么一个合理的假设是，每天收到的邮件数量服从一个'''泊松分布'''。其他可能遵循一个'''泊松分布'''的例子包括：呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。”改为“呼叫中心每小时接到的电话数量和每秒从放射源的衰变数。”

== Definitions定义 ==

===Probability mass function概率分布函数===

The Poisson distribution is popular for modeling the ''number of times an event occurs in an interval of time or space''.

The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space.

'''泊松分布'''模型用来模拟一个事件在一段时间或空间内发生的次数。

A discrete [[random variable]] ''X'' is said to have a Poisson distribution with parameter ''λ'' > 0, if, for ''k'' = 0, 1, 2, ..., the [[probability mass function]] of ''X'' is given by:{{r|Yates2014|p=60}}

A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, ..., the probability mass function of X is given by:

一个离散的随机变量 x 被称为具有参数λ > 0的泊松分布，如果对于 k = 0,1,2，... ，x 的概率分布函数是:

:<math>\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},</math>

where

当

* ''e'' is [[e (mathematical constant)|Euler's number]] (''e'' = 2.71828...)

* ''k''! is the [[factorial]] of ''k''.

The positive [[real number]] ''λ'' is equal to the [[expected value]] of ''X'' and also to its [[variance]]<ref>For the proof, see :

The positive real number λ is equal to the expected value of X and also to its variance<ref>For the proof, see :

正实数'''λ'''等于 x 的期望值和方差相关证明，请参阅:

<ref> [http://www.proofwiki.org/wiki/Expectation_of_Poisson_Distribution Proof wiki: expectation] and [http://www.proofwiki.org/wiki/Variance_of_Poisson_Distribution Proof wiki: variance]</ref>

<ref> [http://www.proofwiki.org/wiki/Expectation_of_Poisson_Distribution Proof wiki: expectation] and [http://www.proofwiki.org/wiki/Variance_of_Poisson_Distribution Proof wiki: variance]</ref>

<ref> [ http://www.proofwiki.org/wiki/expectation_of_poisson_distribution 证明 wiki: 期望]和[ http://www.proofwiki.org/wiki/variance_of_poisson_distribution 证明 wiki: variance ] </ref>

:<math>\lambda=\operatorname{E}(X)=\operatorname{Var}(X).</math>

The Poisson distribution can be applied to systems with a [[large number of rare events|large number of possible events, each of which is rare]]. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.

泊松分布可以应用于包括大量罕见可能事件的系统。在正确的条件下，在一个固定的时间间隔内发生的这类事件的数量是一个具有泊松分布的随机数。

===Example举例===

The Poisson distribution may be useful to model events such as

The Poisson distribution may be useful to model events such as

泊松分布模型可以用来模拟事件，比如

* The number of meteorites greater than 1 meter diameter that strike Earth in a year

* The number of patients arriving in an emergency room between 10 and 11 pm

* The number of laser photons hitting a detector in a particular time interval

* 一年内撞击地球的直径大于1米的陨石数量

* 晚上10点到11点到达急诊室的病人人数

* 在特定时间间隔内撞击探测器的激光光子数

===Assumptions and validity假设与有效条件===

The Poisson distribution is an appropriate model if the following assumptions are true:{{r|Koehrsen2019}}

The Poisson distribution is an appropriate model if the following assumptions are true:

以下假设成立时，泊松分布模型适用:

* {{mvar|k}} is the number of times an event occurs in an interval and {{mvar|k}} can take values 0, 1, 2, ....

* The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.

* The average rate at which events occur is independent of any occurrences. For simplicity, this is usually assumed to be constant, but may in practice vary with time.

* Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur.

* 事件在一个时间间隔内发生且{mvar | k}可以取值0，1，2，...；

* 一个事件的发生不影响第二个事件发生的概率，也就是时间发生相互独立；

* 事件发生的平均速率与任何事件无关。为简单起见，通常假定其为常数，但实际上可能随时间而变化；

* 两个事件不可能在完全相同的时刻发生，即在每一小段的时间内正好有一个事件发生或不发生。

If these conditions are true, then {{mvar|k}} is a Poisson random variable, and the distribution of {{mvar|k}} is a Poisson distribution.

If these conditions are true, then is a Poisson random variable, and the distribution of is a Poisson distribution.

如果这些条件成立，那么它是一个泊松随机变量，其分布是一个''' {泊松分布 Poisson distribution'''。

The Poisson distribution is also the [[Limit (mathematics)|limit]] of a [[binomial distribution]], for which the probability of success for each trial equals {{mvar|λ}} divided by the number of trials, as the number of trials approaches infinity (see [[Poisson distribution#Related distributions|Related distributions]]).

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals divided by the number of trials, as the number of trials approaches infinity (see Related distributions).
每次试验的成功概率除以总试验次数，随着试验的数量趋于无穷大，泊松分布也是''' 二项式分布Binomial distribution'''的极限。（可参考相关分布）

===Probability of events for a Poisson distribution''' {泊松分佈 Poisson distribution'''的事件概率===

An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing {{mvar|k}} events in an interval is given by the equation

An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing events in an interval is given by the equation

一个事件可以在一个间隔内发生0,1,2，... 次。区间内的平均事件数被指定为 < math > lambda </math > (lambda)。Lambda </math > 是'''事件速率Event rate '''，也称为''' 速率参数Rate parameter'''。以下方程给出了在一个区间内观测事件的概率

:<math>P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!}</math>

<math>P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!}</math>

< math > p (k text { events in interval }) = frac { lambda ^ k e ^ {-lambda }{ k！{/math >

where

where

当

* <math> \lambda </math> is the average number of events per interval

* ''e'' is the number 2.71828... ([[e (mathematical constant)|Euler's number]]) the base of the natural logarithms

* {{mvar|k}} takes values 0, 1, 2, ...

* {{mvar|k}}! = {{mvar|k}} × ({{mvar|k}} − 1) × ({{mvar|k}} − 2) × ... × 2 × 1 is the [[factorial]] of {{mvar|k}}.

This equation is the [[probability mass function]] (PMF) for a Poisson distribution.

This equation is the probability mass function (PMF) for a Poisson distribution.

这个方程就是概率质量函数的泊松分布。

This equation can be adapted if, instead of the average number of events <math> \lambda </math>, we are given a time rate <math> r </math> for the events to happen. Then <math> \lambda = r t </math> (with <math> r</math> in units of 1/time), and

This equation can be adapted if, instead of the average number of events <math> \lambda </math>, we are given a time rate <math> r </math> for the events to happen. Then <math> \lambda = r t </math> (with <math> r</math> in units of 1/time), and

如果不用事件的平均数字 < math > lambda </math > ，而是给出事件发生的时间率 < math > r </math > ，那么这个方程就可以适用。然后是 < math > > lambda = r t </math > (以1/time 为单位的 < math > r </math >) ，以及

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“那么这个方程就可以适用”改为“那么这个方程就是适用的”

: <math>P(k \text{ events in interval } t) = \frac{(r t)^k e^{-r t}}{k!}</math>

< math > p (k text { events in interval } t) = frac {(r t) ^ k e ^ {-r t }{ k！{/math >

==== Examples of probability for Poisson distributions''' {泊松分佈 Poisson distribution'''概率的示例 ====

{{col-begin|gap=5em}}

{{col-break}}

On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of {{mvar|k}} = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.

On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.

在某一条河流上，洪水平均每100年发生泛滥一次。计算在100年间洪水泛滥次数= 0,1,2,3,4,5，或6次的概率，假设（其分布）适用泊松模型。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“计算在100年间洪水泛滥次数= 0,1,2,3,4,5或6次的概率”

Because the average event rate is one overflow flood per 100 years, ''λ'' = 1

Because the average event rate is one overflow flood per 100 years, λ = 1

因为平均事件率是每100年发一次洪水，λ = 1

: <math> P(k \text{ overflow floods in 100 years}) = \frac{\lambda^k e^{-\lambda}}{k!} = \frac{1^k e^{-1}}{k!}</math>

<math> P(k \text{ overflow floods in 100 years}) = \frac{\lambda^k e^{-\lambda}}{k!} = \frac{1^k e^{-1}}{k!}</math>

< math > p (k text { overflow in 100 years }) = frac { lambda ^ k e ^ {-lambda }}{ k! }= frac {1 ^ k e ^ {-1}{ k！{/math >

: <math> P(k = 0 \text{ overflow floods in 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{e^{-1}}{1} \approx 0.368 </math>

<math> P(k = 0 \text{ overflow floods in 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{e^{-1}}{1} \approx 0.368 </math>

< math > p (k = 0 text { overflow flood in 100 years }) = frac {1 ^ 0 e ^ {-1}{0! }= frac { e ^ {-1}{1}约0.368 </math >

: <math> P(k = 1 \text{ overflow flood in 100 years}) = \frac{1^1 e^{-1}}{1!} = \frac{e^{-1}}{1} \approx 0.368 </math>

<math> P(k = 1 \text{ overflow flood in 100 years}) = \frac{1^1 e^{-1}}{1!} = \frac{e^{-1}}{1} \approx 0.368 </math>

< math > p (k = 1 text { overflow flood in 100 years }) = frac {1 ^ 1 e ^ {-1}{1! }= frac { e ^ {-1}{1}约0.368 </math >

: <math> P(k = 2 \text{ overflow floods in 100 years}) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2} \approx 0.184 </math>

<math> P(k = 2 \text{ overflow floods in 100 years}) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2} \approx 0.184 </math>

< math > p (k = 2 text { overflow flood in 100 years }) = frac {1 ^ 2 e ^ {-1}{2! }= frac { e ^ {-1}{2} approx 0.184 </math >

{{col-break}}

The table below gives the probability for 0 to 6 overflow floods in a 100-year period.

The table below gives the probability for 0 to 6 overflow floods in a 100-year period.

下表给出了100年内0到6次洪水泛滥的概率。

{| class="wikitable"

{| class="wikitable"

{ | class = “ wikitable”

|-

|-

|-

! {{mvar|k}} !! ''P''({{mvar|k}} overflow floods in 100 years)

! !! P( overflow floods in 100 years)

!!!100年内泛滥成灾

|-

|-

|-

| 0|| 0.368

| 0|| 0.368

| 0|| 0.368

|-

|-

|-

| 1|| 0.368

| 1|| 0.368

| 1|| 0.368

|-

|-

|-

| 2|| 0.184

| 2|| 0.184

| 2|| 0.184

|-

|-

|-

| 3|| 0.061

| 3|| 0.061

| 3|| 0.061

|-

|-

|-

| 4|| 0.015

| 4|| 0.015

| 4|| 0.015

|-

|-

|-

| 5|| 0.003

| 5|| 0.003

| 5|| 0.003

|-

|-

|-

| 6|| 0.0005

| 6|| 0.0005

| 6|| 0.0005

|}

|}

|}

{{col-end}}

{{col-begin|gap=5em}}

{{col-break}}

Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.{{r|Ugarte2016}}

Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.

乌加特和他的同事们在一篇报道中提到，世界杯足球赛的平均进球数约为2.5个，这也适用泊松模型。

Because the average event rate is 2.5 goals per match, ''λ'' = 2.5.

Because the average event rate is 2.5 goals per match, λ = 2.5.

因为平均每场比赛有2.5个进球，λ = 2.5个。

: <math> P(k \text{ goals in a match}) = \frac{2.5^k e^{-2.5}}{k!}</math>

<math> P(k \text{ goals in a match}) = \frac{2.5^k e^{-2.5}}{k!}</math>

< math > p (k text { goals in a match }) = frac {2.5 ^ k e ^ {-2.5}{ k！{/math >

: <math> P(k = 0 \text{ goals in a match}) = \frac{2.5^0 e^{-2.5}}{0!} = \frac{e^{-2.5}}{1} \approx 0.082 </math>

<math> P(k = 0 \text{ goals in a match}) = \frac{2.5^0 e^{-2.5}}{0!} = \frac{e^{-2.5}}{1} \approx 0.082 </math>

= frac {2.5 ^ 0 e ^ {-2.5}{0! }= frac { e ^ {-2.5}{1}大约0.082

: <math> P(k = 1 \text{ goal in a match}) = \frac{2.5^1 e^{-2.5}}{1!} = \frac{2.5 e^{-2.5}}{1} \approx 0.205 </math>

<math> P(k = 1 \text{ goal in a match}) = \frac{2.5^1 e^{-2.5}}{1!} = \frac{2.5 e^{-2.5}}{1} \approx 0.205 </math>

= frac {2.5 ^ 1 e ^ {-2.5}{1! }= frac {2.5 e ^ {-2.5}{1}约0.205

: <math> P(k = 2 \text{ goals in a match}) = \frac{2.5^2 e^{-2.5}}{2!} = \frac{6.25 e^{-2.5}}{2} \approx 0.257 </math>

<math> P(k = 2 \text{ goals in a match}) = \frac{2.5^2 e^{-2.5}}{2!} = \frac{6.25 e^{-2.5}}{2} \approx 0.257 </math>

< math > p (k = 2 text { goals in a match }) = frac {2.5 ^ 2 e ^ {-2.5}{2! }= frac {6.25 e ^ {-2.5}{2}{2}{约0.257 </math >

{{col-break}}

The table below gives the probability for 0 to 7 goals in a match.

The table below gives the probability for 0 to 7 goals in a match.

下表给出了一场比赛中0到7个进球的概率。

{| class="wikitable"

{| class="wikitable"

{ | class = “ wikitable”

|-

|-

|-

! {{mvar|k}} !! ''P''({{mvar|k}} goals in a World Cup soccer match)

! !! P( goals in a World Cup soccer match)

!!!P (世界杯足球赛进球)

|-

|-

|-

| 0|| 0.082

| 0|| 0.082

| 0|| 0.082

|-

|-

|-

| 1|| 0.205

| 1|| 0.205

| 1|| 0.205

|-

|-

|-

| 2|| 0.257

| 2|| 0.257

| 2|| 0.257

|-

|-

|-

| 3|| 0.213

| 3|| 0.213

| 3|| 0.213

|-

|-

|-

| 4|| 0.133

| 4|| 0.133

| 4|| 0.133

|-

|-

|-

| 5|| 0.067

| 5|| 0.067

| 5|| 0.067

|-

|-

|-

| 6|| 0.028

| 6|| 0.028

| 6|| 0.028

|-

|-

|-

| 7|| 0.010

| 7|| 0.010

| 7|| 0.010

|}

|}

|}

{{col-end}}

====Once in an interval events: The special case of ''λ'' = 1 and ''k'' = 0 事件唯一发生：''λ'' = 1 与 ''k'' = 0的特殊情形====

Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years (''λ'' = 1 event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of {{mvar|k}} = 0 meteorite hits in the next 100 years?

Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years (λ = 1 event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of = 0 meteorite hits in the next 100 years?

假设天文学家估计，大型陨石(超过一定大小)平均每100年撞击地球一次(= 每100年撞击一次) ，而且陨石撞击的次数服从泊松分布。在接下来的100年里，被陨石击中k=0的概率是多少？

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“而且陨石撞击的次数紧随泊松分布之后”改为“而且陨石撞击的次数紧随泊松分布之后”

: <math> P(k = \text{0 meteorites hit in next 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{1}{e} \approx 0.37 </math>

<math> P(k = \text{0 meteorites hit in next 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{1}{e} \approx 0.37 </math>

< math > p (k = text {0 meteorites hit in next 100 years }) = frac {1 ^ 0 e ^ {-1}{0! }= frac {1}{ e }大约0.37 </math >

Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1 − 0.37 = 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years.

Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1 − 0.37 = 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years.

根据这些假设，未来100年内没有大陨石撞击地球的概率大约为0.37。剩下的1-0.37 = 0.63是未来100年内被1,2,3或更多大型陨石撞击的概率。

In an example above, an overflow flood occurred once every 100 years (''λ'' = 1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.

In an example above, an overflow flood occurred once every 100 years (λ = 1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.

在上面的一个例子中，洪水每100年发生泛滥一次(''λ''= 1)。根据同样的计算，100年内不会有洪水泛滥的概率大约是0.37。

In general, if an event occurs on average once per interval (''λ'' = 1), and the events follow a Poisson distribution, then {{nowrap|1=''P''(0 events in next interval) = 0.37}}. In addition, ''P''(exactly one event in next interval) = 0.37, as shown in the table for overflow floods.

In general, if an event occurs on average once per interval (λ = 1), and the events follow a Poisson distribution, then . In addition, P(exactly one event in next interval) = 0.37, as shown in the table for overflow floods.

一般来说，如果一个事件平均每个时间间隔发生一次(''λ''= 1) ，并且事件遵循泊松分布，那么p (下一个间隔中正好有一个事件) = 0.37，如洪水泛滥的表所示。

=== Examples that violate the Poisson assumptions 违反泊松假设的例子===

The number of students who arrive at the [[Student center|student union]] per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).

The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).

每分钟抵达学生会的学生人数可能不会遵循一个''' 泊松分佈Poisson distribution.'''，因为这个比率不是恒定的(上课时间的低比率，课间时的高比率) ，而且每个学生的到达也不是独立的(学生往往是成群结队来的)。

The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.

The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.

一次大的强震会增加发生类似震级余震的可能性，那么一个国家每年发生5级地震的次数可能不会服从''' 泊松分佈Poisson distribution.'''。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a [[Zero-truncated Poisson distribution]].

Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a Zero-truncated Poisson distribution.

至少有一个事件确定发生的情况不是 Poission 分布式的，但也许可以使用零截断''' 泊松分佈Poisson distribution.'''进行建模。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a [[Zero-inflated model]].

Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a Zero-inflated model.

如果零事件的区间数高于泊松模型预测的区间数分布，则可以使用零膨胀模型来建模。

== Properties 性能==

=== ''' 描述统计学Descriptive statistics'''===

* The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.

*一个泊松分布随机变量的期望值和方差均等于λ

* The [[coefficient of variation]] is <math>\textstyle \lambda^{-1/2}</math>, while the [[index of dispersion]] is 1.{{r|Johnson2005|p=163}}

*当[[index of dispersion]] is 1.{{r|Johnson2005|p=163}}，相关系数为<math>\textstyle \lambda^{-1/2}</math>

* The [[mean absolute deviation]] about the mean is{{r|Johnson2005|p=163}}

*均值的绝对平均差为{{r|Johnson2005|p=163}}

::<math>\operatorname{E}[|X-\lambda|]= \frac{2 \lambda^{\lfloor\lambda\rfloor + 1} e^{-\lambda}}{\lfloor\lambda\rfloor!}.</math>

<math>\operatorname{E}[|X-\lambda|]= \frac{2 \lambda^{\lfloor\lambda\rfloor + 1} e^{-\lambda}}{\lfloor\lambda\rfloor!}.</math>

[ | x-lambda | ] = frac {2 lambda ^ { lfloor lambda rfloor + 1} e ^ {-lambda }{ lfloor lambda rfloor！} . </math >

* The [[mode (statistics)|mode]] of a Poisson-distributed random variable with non-integer λ is equal to <math>\scriptstyle\lfloor \lambda \rfloor</math>, which is the largest integer less than or equal to ''λ''. This is also written as [[floor function|floor]](λ). When λ is a positive integer, the modes are ''λ'' and ''λ'' − 1.

* All of the [[cumulant]]s of the Poisson distribution are equal to the expected value ''λ''. The ''n''th [[factorial moment]] of the Poisson distribution is ''λ''''n''.

* The [[expected value]] of a [[Poisson process]] is sometimes decomposed into the product of ''intensity'' and ''exposure'' (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as “exposure”).{{r|Helske2017}}

*一个具有非整值λ 的泊松分布随机变量的统计值等于<math>\scriptstyle\lfloor \lambda \rfloor</math>, 小于''λ''的最大整数。它也写作[[floor function|floor]](λ). λ为正整数时，取值为''λ'' 以及 ''λ'' − 1。
*所有泊松分布的积均等于期望值 ''λ''。泊松分布的n阶指数积为''λ''''n''。
*期望值与泊松过程有时分解为“强度”与“面积”的乘积（或更一般地表示为强度函数随时间或空间的积分，有时描述为'''“暴露”“exposure”'''。）{{r|Helske2017}}

=== Median 中值===

Bounds for the median (<math>\nu</math>) of the distribution are known and are [[Mathematical jargon#sharp|sharp]]:{{r|Choi1994}}

Bounds for the median (<math>\nu</math>) of the distribution are known and are sharp:

分布的中位数(< math > nu </math >)的界限为已知，且清晰:

: <math> \lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}. </math>

<math> \lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}. </math>

2 le nu < lambda + frac {1}{3}.数学

=== Higher moments 高阶矩===

* The higher [[moment (mathematics)|moments]] ''m''''k'' of the Poisson distribution about the origin are [[touchard polynomials]] in λ:

:: <math> m_k = \sum_{i=0}^k \lambda^i \left\{\begin{matrix} k \\ i \end{matrix}\right\},</math>

<math> m_k = \sum_{i=0}^k \lambda^i \left\{\begin{matrix} k \\ i \end{matrix}\right\},</math>

< math > m _ k = sum { i = 0} ^ k lambda ^ i left { begin { matrix } k i end { matrix } right } ，</math >

*泊松分布原点的高阶矩[[moment (mathematics)|moments]] ''m''''k''是同余多项式，λ中：

: where the {braces} denote [[Stirling numbers of the second kind]].{{r|Riordan1937}}{{r|Haight1967|p=6}} The coefficients of the polynomials have a [[combinatorics|combinatorial]] meaning. In fact, when the expected value of the Poisson distribution is 1, then [[Dobinski's formula]] says that the ''n''th moment equals the number of [[partition of a set|partitions of a set]] of size ''n''.

where the {braces} denote Stirling numbers of the second kind. The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth moment equals the number of partitions of a set of size n.

其中{括号}表示第二类 Stirling 数。多项式的系数具有组合意义。事实上，当泊松分佈的期望值是1时，那么 Dobinski 的公式说第 n 个时刻等于一组大小为 n 的分区的数目。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

For the non-centered moments we define <math>B=k/\lambda</math>, then{{r|Jagadeesan2017}}

For the non-centered moments we define <math>B=k/\lambda</math>, then

对于非中心时刻，我们定义了 < math > b = k/lambda </math >

::<math>

<math>

《数学》

E[X^k]^{1/k} \le C\cdot

E[X^k]^{1/k} \le C\cdot

E [ x ^ k ] ^ {1/k } le c dot

\begin{cases}

\begin{cases}

开始{ cases }

k/B & \text{if}\quad B < e \\

k/B & \text{if}\quad B < e \\

k/B & text { if }方 b < e

k/\log B & \text{if}\quad B\ge e

k/\log B & \text{if}\quad B\ge e

如果你想要一个更好的工具，你需要一个更好的工具

\end{cases}

\end{cases}

结束{ cases }

</math>

</math>

数学

where <math>C</math> is some absolute constant greater than 0.

where <math>C</math> is some absolute constant greater than 0.

其中，c </math > 是某个大于0的绝对常数。

=== Sums of Poisson-distributed random variables 泊松分布随机变量和===

: If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are [[statistical independence|independent]], then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>.{{r|Lehmann1986|p=65}} A converse is [[Raikov's theorem]], which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.{{r|Raikov1937}}{{r|vonMises1964|p=}}

If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are independent, then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>. A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.

如果对于 < math > i = 1，dotsc，n </math > 是独立的，那么 < math > sum { i = 1} ^ n xi sim 操作者名{ Pois }左(sum { i = 1} ^ n lambda _ i 右) </math > 。一个逆定理是雷科夫定理，它说如果两个独立的随机变量之和是''' 泊松分佈Poisson distribution.'''的，那么这两个独立的随机变量之和也是''' 泊松分佈Poisson distribution.'''的。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

=== Other properties 其他特性===

* The Poisson distributions are [[Infinite divisibility (probability)|infinitely divisible]] probability distributions.{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}

* The directed [[Kullback–Leibler divergence]] of <math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> is given by

*泊松分布是[[无限可除性（概率）|无限可除]] 概率分布，{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}。
*<math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> 的直接''' 相对熵（K-L散度）[[Kullback–Leibler divergence]]'''由以下给出：

:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>

<math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>

< math > operatorname { d }{ text { KL }}(lambda mid lambda _ 0) = lambda _ 0-lambda + lambda log frac { lambda }{ lambda _ 0} . </math >

* Bounds for the tail probabilities of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> can be derived using a [[Chernoff bound]] argument.{{r|Mitzenmacher2005|p=97-98}}
*泊松随机变量尾概率的界<math> X \sim \operatorname{Pois}(\lambda)</math> 可以用[[''' 切诺夫界Chernoff bound''']]参数派生{{r|Mitzenmacher2005|p=97-98}}
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,

<math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,

(x x x) leq frac {(e lambda) ^ x e ^ {-lambda }{ x ^ x } ，text { for } x > lambda </math > ,

:: <math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math>

<math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math>

< math > p (x leq x) leq frac {(e lambda) ^ x e ^ {-lambda }{ x ^ x } ，text { for } x < lambda. </math >

* The upper tail probability can be tightened (by a factor of at least two) as follows:{{r|Short2013}}

*长尾概率可被收紧（至少两倍）如下：{{r|Short2013}}
:: <math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(x\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(x\mid\lambda)}})}, \text{ for } x > \lambda,</math>

<math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(x\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(x\mid\lambda)}})}, \text{ for } x > \lambda,</math>

< math > p (x geq x) leq frac { e ^ {-operatorname { d }{ text { KL }(x mid lambda)}}{ max {(2，sqrt {4 pi operatorname { d }{ text { KL }(x mid lambda)}}}}) ，text { for } x > lambda，</math >

: where <math>\operatorname{D}_{\text{KL}}(x\mid\lambda)</math> is the directed Kullback–Leibler divergence, as described above.

where <math>\operatorname{D}_{\text{KL}}(x\mid\lambda)</math> is the directed Kullback–Leibler divergence, as described above.

其中 < math > operatorname { d } _ { text { KL }}(x mid lambda) </math > 是指向的 Kullback-Leibler 分歧，如上所述。

* Inequalities that relate the distribution function of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> to the [[Standard normal distribution]] function <math> \Phi(x) </math> are as follows:{{r|Short2013}}

*关于泊松随机变量分布函数的不等式 <math> X \sim \operatorname{Pois}(\lambda)</math>与标准正态分布函数<math> \Phi(x) </math> 如下:{{r|Short2013}}

:: <math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math>

<math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math>

< math > Phi left (operatorname { sign }(k-lambda) sqrt {2 operatorname { d }{ text { KL }(k mid lambda)}右) 0，</math >

: where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is again the directed Kullback–Leibler divergence.

where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is again the directed Kullback–Leibler divergence.

其中 < math > 操作者名{ d } _ { text { KL }(k mid lambda) </math > 仍然是有向的 Kullback-Leibler 分歧。

=== Poisson races 泊松族群===

Let <math>X \sim \operatorname{Pois}(\lambda)</math> and <math>Y \sim \operatorname{Pois}(\mu)</math> be independent random variables, with <math> \lambda < \mu </math>, then we have that

Let <math>X \sim \operatorname{Pois}(\lambda)</math> and <math>Y \sim \operatorname{Pois}(\mu)</math> be independent random variables, with <math> \lambda < \mu </math>, then we have that

设 x sim 操作者名称{ Pois }(lambda) </math > 和 < math > y sim 操作者名称{ Pois }(mu) </math > 是独立的随机变量，并且带有 < math > lambda < mu </math > ，那么我们就有了

:<math>

<math>

《数学》

\frac{e^{-(\sqrt{\mu} -\sqrt{\lambda})^2 }}{(\lambda + \mu)^2} - \frac{e^{-(\lambda + \mu)}}{2\sqrt{\lambda \mu}} - \frac{e^{-(\lambda + \mu)}}{4\lambda \mu} \leq P(X - Y \geq 0) \leq e^{- (\sqrt{\mu} -\sqrt{\lambda})^2}

\frac{e^{-(\sqrt{\mu} -\sqrt{\lambda})^2 }}{(\lambda + \mu)^2} - \frac{e^{-(\lambda + \mu)}}{2\sqrt{\lambda \mu}} - \frac{e^{-(\lambda + \mu)}}{4\lambda \mu} \leq P(X - Y \geq 0) \leq e^{- (\sqrt{\mu} -\sqrt{\lambda})^2}

Frac { e ^ {-(sqrt { mu }-sqrt { lambda }) ^ 2}{(lambda + mu) ^ 2}-frac { e ^ {-(lambda + mu)}}{2 sqrt { lambda mu }-frac { e ^ { e ^ {-(lambda + mu)}-(lambda + mu)}}}{4 lambda } leq p (x-y geq 0) leq ^ e ^ {-(sqrt { mu }-sqrt { lambda }) ^ 2}

</math>

</math>

数学

The upper bound is proved using a standard Chernoff bound.

The upper bound is proved using a standard Chernoff bound.

利用标准的''' 切诺夫界Chernoff bound'''证明了上界的存在性。

The lower bound can be proved by noting that <math> P(X-Y\geq0\mid X+Y=i)</math> is the probability that <math>Z \geq \frac{i}{2}</math>, where <math>Z \sim \operatorname{Bin}\left(i, \frac{\lambda}{\lambda+\mu}\right)</math>, which is bounded below by <math> \frac{1}{(i+1)^2} e^{\left(-iD\left(0.5 \| \frac{\lambda}{\lambda+\mu}\right)\right)} </math>, where <math>D</math> is [[Kullback–Leibler divergence|relative entropy]] (See the entry on [[Binomial distribution#Tail Bounds|bounds on tails of binomial distributions]] for details). Further noting that <math> X+Y \sim \operatorname{Pois}(\lambda+\mu)</math>, and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath ''et al.''.{{r|Kamath2015}}

The lower bound can be proved by noting that <math> P(X-Y\geq0\mid X+Y=i)</math> is the probability that <math>Z \geq \frac{i}{2}</math>, where <math>Z \sim \operatorname{Bin}\left(i, \frac{\lambda}{\lambda+\mu}\right)</math>, which is bounded below by <math> \frac{1}{(i+1)^2} e^{\left(-iD\left(0.5 \| \frac{\lambda}{\lambda+\mu}\right)\right)} </math>, where <math>D</math> is relative entropy (See the entry on bounds on tails of binomial distributions for details). Further noting that <math> X+Y \sim \operatorname{Pois}(\lambda+\mu)</math>, and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al..

下限可以通过下面的例子来证明: < math > p (X-Y geq0 mid x + y = i) </math > 是 < math > z geq frac { i }{2} </math > ，其中 < math > z sim 操作员名{ Bin }左(i，frac { lambda } + mu }右) </math > ,下面由 < math > frac {1}{(i + 1) ^ 2} e ^ { left (- iD left (0.5 | frac { lambda }(- iD left (0.5 | frac { lambda + mu } right)}}} </math > 限定，其中 < math > d </math > 是相对熵。进一步注意到 < math > x + y sim 操作者名{ Pois }(lambda + mu) </math > ，并计算无条件概率的下限得到结果。更多的细节可以在卡马斯等人的附录中找到。

== ''' 相关分布'Related distributions''==

===Genera通常l===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“Genera通常l”改为“General通常”
* If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the difference <math> Y = X_1 - X_2</math> follows a [[Skellam distribution]].

* If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the distribution of <math>X_1</math> conditional on <math>X_1+X_2</math> is a [[binomial distribution]].

:Specifically, if <math>X_1+X_2=k</math>, then <math>\!X_1\sim \mathrm{Binom}(k, \lambda_1/(\lambda_1+\lambda_2))</math>.

Specifically, if <math>X_1+X_2=k</math>, then <math>\!X_1\sim \mathrm{Binom}(k, \lambda_1/(\lambda_1+\lambda_2))</math>.

具体来说，如果 < math > x _ 1 + x _ 2 = k </math > ，那么 < math > ! x _ 1 sim mathrm { Binom }(k，lambda _ 1/(lambda _ 1 + lambda _ 2)) </math > 。

:More generally, if ''X''1, ''X''2,..., ''X''''n'' are independent Poisson random variables with parameters ''λ''1, ''λ''2,..., ''λ''''n'' then

More generally, if X1, X2,..., Xn are independent Poisson random variables with parameters λ1, λ2,..., λn then

更一般地说，如果 x 1 ，x 2 ，... ，x n 是独立的随机变量，参数 1 ， 2 ，... ， n 然后

:: given <math>\sum_{j=1}^n X_j=k,</math> <math>X_i \sim \mathrm{Binom}\left(k, \frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right)</math>. In fact, <math>\{X_i\} \sim \mathrm{Multinom}\left(k, \left\{\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right\}\right)</math>.

given <math>\sum_{j=1}^n X_j=k,</math> <math>X_i \sim \mathrm{Binom}\left(k, \frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right)</math>. In fact, <math>\{X_i\} \sim \mathrm{Multinom}\left(k, \left\{\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right\}\right)</math>.

给定的数学公式 sum { j = 1} ^ n x _ j = k，</math > x _ i sim mathrm { Binom }左(k，frac { lambda _ i }{ sum { j = 1} ^ n lambda _ j }右) </math > 。事实上，[ math ]{ xi } sim mathrm { mothom } left (k，left { frac { lambda _ i }{ sum { j = 1} ^ n lambda _ j } right }}) </math > 。

* If <math>X \sim \mathrm{Pois}(\lambda)\,</math> and the distribution of <math>Y</math>, conditional on ''X'' = ''k'', is a [[binomial distribution]], <math>Y \mid (X = k) \sim \mathrm{Binom}(k, p)</math>, then the distribution of Y follows a Poisson distribution <math>Y \sim \mathrm{Pois}(\lambda \cdot p)\,</math>. In fact, if <math>\{Y_i\} </math>, conditional on X = k, follows a multinomial distribution, <math>\{Y_i\} \mid (X = k) \sim \mathrm{Multinom}\left(k, p_i\right)</math>, then each <math>Y_i</math> follows an independent Poisson distribution <math>Y_i \sim \mathrm{Pois}(\lambda \cdot p_i), \rho(Y_i, Y_j) = 0</math>.

* The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the [[expected value|expected]] number of successes remains fixed — see [[#law of rare events|law of rare events]] below. Therefore, it can be used as an approximation of the binomial distribution if ''n'' is sufficiently large and ''p'' is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and ''p'' is smaller than or equal to 0.05, and an excellent approximation if ''n'' ≥ 100 and ''np'' ≤ 10.{{r|NIST2006}}

:: <math>F_\mathrm{Binomial}(k;n, p) \approx F_\mathrm{Poisson}(k;\lambda=np)\,</math>

<math>F_\mathrm{Binomial}(k;n, p) \approx F_\mathrm{Poisson}(k;\lambda=np)\,</math>

(k; n，p)接近 f _ mathrm { Poisson }(k; lambda = np) ，</math >

* The Poisson distribution is a [[special case]] of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.{{r|Zhang2013|Zhang2016}} The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a [[compound Poisson distribution#Special cases|special case]] of a [[compound Poisson distribution]].
*这一泊松分布是离散复合泊松分布（或断续泊松分布）在只有一个参数情况下的[[特殊情形]] 。{{r|Zhang2013|Zhang2016}}离散复合泊松分布可由一元多项式分布的极限分布导出。同时它也是[[复合泊松分布#特殊情况]] 复合泊松分布的一个特例。

* For sufficiently large values of λ, (say λ>1000), the [[normal distribution]] with mean λ and variance λ (standard deviation <math>\sqrt{\lambda}</math>) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [[continuity correction]] is performed, i.e., if P(''X'' ≤ ''x''), where ''x'' is a non-negative integer, is replaced by P(''X'' ≤ ''x'' + 0.5).
*对于足够大的值λ，（如 λ>1000)，具有均值 λ 的正态分布与变量 λ （标准差 <math>\sqrt{\lambda}</math>），是泊松分布的完美近似。如果 λ 大于10，则正态分布在适当的校正下可近似模拟，例如如果P(''X'' ≤ ''x'')，''x'' 为非负整数，则将其改为P(''X'' ≤ ''x'' + 0.5)。

:: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>

<math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>

(x; λ) approx f _ mathrm { normal }(x; mu = lambda，sigma ^ 2 = lambda) ，</math >

* [[Variance-stabilizing transformation]]: If <math>X \sim \mathrm{Pois}(\lambda)\,</math>, then

::<math>Y = 2 \sqrt{X} \approx \mathcal{N}(2\sqrt{\lambda};1)</math>,{{r|Johnson2005|p=168}}

<math>Y = 2 \sqrt{X} \approx \mathcal{N}(2\sqrt{\lambda};1)</math>,

(2 sqrt { lambda } ; 1) </math > ,

:and

and

及

::<math>Y = \sqrt{X} \approx \mathcal{N}(\sqrt{\lambda};1/4)</math>.{{r|McCullagh1989|p=196}}

<math>Y = \sqrt{X} \approx \mathcal{N}(\sqrt{\lambda};1/4)</math>.

(sqrt { lambda } ; 1/4) </math > .

:Under this transformation, the convergence to normality (as <math>\lambda</math> increases) is far faster than the untransformed variable.{{Citation needed|date=May 2012}} Other, slightly more complicated, variance stabilizing transformations are available,{{r|Johnson2005|p=168}} one of which is [[Anscombe transform]].{{r|Anscombe1948}} See [[Data transformation (statistics)]] for more general uses of transformations.

Under this transformation, the convergence to normality (as <math>\lambda</math> increases) is far faster than the untransformed variable. Other, slightly more complicated, variance stabilizing transformations are available, one of which is Anscombe transform. See Data transformation (statistics) for more general uses of transformations.

在这种转换下，收敛到正态的速度(如 < math > lambda </math > 增加)远远快于未转换的变量。还有一些稍微复杂一些的稳定方差的变换，其中之一就是安斯科姆变换。有关转换的更多一般用途，请参见数据转换(统计信息)。

* If for every ''t'' > 0 the number of arrivals in the time interval [0, ''t''] follows the Poisson distribution with mean ''λt'', then the sequence of inter-arrival times are independent and identically distributed [[exponential distribution|exponential]] random variables having mean 1/''λ''.{{r|Ross2010|p=317–319}}

*

* The [[cumulative distribution function]]s of the Poisson and [[chi-squared distribution]]s are related in the following ways:{{r|Johnson2005|p=167}}

::<math>F_\text{Poisson}(k;\lambda) = 1-F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,</math>

<math>F_\text{Poisson}(k;\lambda) = 1-F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,</math>

1-F _ { chi ^ 2}(2 lambda; 2(k + 1)) quad text { integer } k，</math >

:and{{r|Johnson2005|p=158}}

and

及

::<math>\Pr(X=k)=F_{\chi^2}(2\lambda;2(k+1)) -F_{\chi^2}(2\lambda;2k) .

<math>\Pr(X=k)=F_{\chi^2}(2\lambda;2(k+1)) -F_{\chi^2}(2\lambda;2k) .

< math > Pr (x = k) = f { chi ^ 2}(2 lambda; 2(k + 1))-f { chi ^ 2}(2 lambda; 2k).

</math>

</math>

数学

===''' Poisson Approximation 泊松近似'''===

Assume <math>X_1\sim\operatorname{Pois}(\lambda_1), X_2\sim\operatorname{Pois}(\lambda_2), \dots, X_n\sim\operatorname{Pois}(\lambda_n)</math> where <math>\lambda_1 + \lambda_2 + \dots + \lambda_n=1</math>, then<ref>{{Cite web | url=https://newonlinecourses.science.psu.edu/stat504/node/48/ | title=1.7.7 – Relationship between the Multinomial and Poisson | STAT 504}}</ref> <math>(X_1, X_2, \dots, X_n)</math> is [[Multinomial distribution|multinomially distributed]]

Assume <math>X_1\sim\operatorname{Pois}(\lambda_1), X_2\sim\operatorname{Pois}(\lambda_2), \dots, X_n\sim\operatorname{Pois}(\lambda_n)</math> where <math>\lambda_1 + \lambda_2 + \dots + \lambda_n=1</math>, then <math>(X_1, X_2, \dots, X_n)</math> is multinomially distributed

假设 x _ 1 sim 操作者名为{ Pois }(lambda _ 1) ，x _ 2 sim 操作者名为{ Pois }(lambda _ 2) ，点，xn sim 操作者名为{ Pois }(lambda _ n) </math > 其中 < math > lambda _ 1 + lambda _ 2 + dots + lambda _ n = 1 </math 多项式 > ，那么 < math > (x _ 1，x _ 2，dots，x _ n) </math > 是统一分布的

<math>(X_1, X_2, \dots, X_n) \sim \operatorname{Mult}(N, \lambda_1, \lambda_2, \dots, \lambda_n)</math> conditioned on <math>N = X_1 + X_2 + \dots X_n</math>.

<math>(X_1, X_2, \dots, X_n) \sim \operatorname{Mult}(N, \lambda_1, \lambda_2, \dots, \lambda_n)</math> conditioned on <math>N = X_1 + X_2 + \dots X_n</math>.

(x _ 1，x _ 2，dots，x _ n) sim 操作员名称{ Mult }(n，λ _ 1，λ _ 2，dots，λ _ n) </math > 取决于 < math > n = x1 + x _ 2 + dots x _ n </math > 。

This means{{r|Mitzenmacher2005|p=101-102}}, among other things, that for any nonnegative function <math>f(x_1,x_2,\dots,x_n)</math>,

This means, among other things, that for any nonnegative function <math>f(x_1,x_2,\dots,x_n)</math>,

这意味着，对于任何非负函数 f (x _ 1，x _ 2，dots，x _ n) </math > ,

if <math>(Y_1, Y_2, \dots, Y_n)\sim\operatorname{Mult}(m, \mathbf{p})</math> is multinomially distributed, then

if <math>(Y_1, Y_2, \dots, Y_n)\sim\operatorname{Mult}(m, \mathbf{p})</math> is multinomially distributed, then

如果[ math ](y _ 1，y _ 2，dots，y _ n) sim 操作符名称{ Mult }(m，mathbf { p }) </math > 是多项式分布，则

:<math>

<math>

《数学》

\operatorname{E}[f(Y_1, Y_2, \dots, Y_n)] \le e\sqrt{m}\operatorname{E}[f(X_1, X_2, \dots, X_n)]

\operatorname{E}[f(Y_1, Y_2, \dots, Y_n)] \le e\sqrt{m}\operatorname{E}[f(X_1, X_2, \dots, X_n)]

操作员名称{ e }[ f (y _ 1，y _ 2，dots，y _ n)] le e sqrt { m }操作员名称{ e }[ f (x _ 1，x _ 2，dots，x _ n)]

</math>

</math>

数学

where <math>(X_1, X_2, \dots, X_n)\sim\operatorname{Pois}(\mathbf{p})</math>.

where <math>(X_1, X_2, \dots, X_n)\sim\operatorname{Pois}(\mathbf{p})</math>.

其中 < math > (x _ 1，x _ 2，dots，x _ n) sim 操作员名称{ Pois }(mathbf { p }) </math > 。

The factor of <math>e\sqrt{m}</math> can be removed if <math>f</math> is further assumed to be monotonically increasing or decreasing.

The factor of <math>e\sqrt{m}</math> can be removed if <math>f</math> is further assumed to be monotonically increasing or decreasing.

如果进一步假定 < math > f </math > 是单调递增或递减的，则可以去掉 < math > e sqrt { m } </math > 的因子。

=== ''' 二元泊松分布Bivariate Poisson distribution'''===

This distribution has been extended to the [[bivariate]] case.{{r|Loukas1986}} The [[generating function]] for this distribution is

This distribution has been extended to the bivariate case. The generating function for this distribution is

这种分布已经扩展到二元情况。这个分布的母函数是

: <math> g( u, v ) = \exp[ ( \theta_1 - \theta_{12} )( u - 1 ) + ( \theta_2 - \theta_{12} )(v - 1) + \theta_{12} ( uv - 1 ) ] </math>

<math> g( u, v ) = \exp[ ( \theta_1 - \theta_{12} )( u - 1 ) + ( \theta_2 - \theta_{12} )(v - 1) + \theta_{12} ( uv - 1 ) ] </math>

< math > g (u，v) = exp [(theta _ 1-theta _ {12})(u-1) + (theta _ 2-theta _ {12})(v-1) + theta _ {12}(uv-1)] </math >

with

with

与

: <math> \theta_1, \theta_2 > \theta_{ 12 } > 0 \, </math>

<math> \theta_1, \theta_2 > \theta_{ 12 } > 0 \, </math>

1，theta 2，theta {12} > 0，</math >

The marginal distributions are Poisson(''θ''1) and Poisson(''θ''2) and the correlation coefficient is limited to the range

The marginal distributions are Poisson(θ1) and Poisson(θ2) and the correlation coefficient is limited to the range

边缘分布为 Poisson ( 1 )和 Poisson ( 2 ) ，相关系数仅限于一定范围

: <math> 0 \le \rho \le \min\left\{ \frac{ \theta_1 }{ \theta_2 }, \frac{ \theta_2 }{ \theta_1 } \right\}</math>

<math> 0 \le \rho \le \min\left\{ \frac{ \theta_1 }{ \theta_2 }, \frac{ \theta_2 }{ \theta_1 } \right\}</math>

[数学][数学][数学][数学]

A simple way to generate a bivariate Poisson distribution <math>X_1,X_2</math> is to take three independent Poisson distributions <math>Y_1,Y_2,Y_3</math> with means <math>\lambda_1,\lambda_2,\lambda_3</math> and then set <math>X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3</math>. The probability function of the bivariate Poisson distribution is

A simple way to generate a bivariate Poisson distribution <math>X_1,X_2</math> is to take three independent Poisson distributions <math>Y_1,Y_2,Y_3</math> with means <math>\lambda_1,\lambda_2,\lambda_3</math> and then set <math>X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3</math>. The probability function of the bivariate Poisson distribution is

一个简单的方法来产生一个二变量的泊松分佈分布: 取3个独立的 Poisson 分布 < math > y _ 1，y _ 2，y _ 3 </math > ，用 < math > lambda _ 1，lambda _ 2，lambda _ 3 </math > 然后设置 < math > x _ 1 = y _ 1 + y _ 3，x _ 2 = y _ 2 + y _ 3 </math > 。二元概率密度函数变量的泊松分佈变量是
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈分布”改为“泊松分布”，“泊松分佈变量”改为“泊松分布变量”
: <math>

<math>

《数学》

\begin{align}

\begin{align}

开始{ align }

& \Pr(X_1=k_1,X_2=k_2) \\

& \Pr(X_1=k_1,X_2=k_2) \\

& Pr (x _ 1 = k _ 1，x _ 2 = k _ 2)

= {} & \exp\left(-\lambda_1-\lambda_2-\lambda_3\right) \frac{\lambda_1^{k_1}}{k_1!} \frac{\lambda_2^{k_2}}{k_2!} \sum_{k=0}^{\min(k_1,k_2)} \binom{k_1}{k} \binom{k_2}{k} k! \left( \frac{\lambda_3}{\lambda_1\lambda_2}\right)^k

= {} & \exp\left(-\lambda_1-\lambda_2-\lambda_3\right) \frac{\lambda_1^{k_1}}{k_1!} \frac{\lambda_2^{k_2}}{k_2!} \sum_{k=0}^{\min(k_1,k_2)} \binom{k_1}{k} \binom{k_2}{k} k! \left( \frac{\lambda_3}{\lambda_1\lambda_2}\right)^k

= {} & exp left (- lambda _ 1-lambda _ 2-lambda _ 3 right) frac { lambda _ 1 ^ { k _ 1}{ k _ 1! }2 ^ { k2}{ k2! }{ k = 0} ^ { min (k _ 1，k _ 2)} binom { k _ 1}{ k } binom { k _ 2}{ k } ！左(frac { lambda _ 3}{ lambda _ 1 lambda _ 2}右) ^ k

\end{align}

\end{align}

结束{ align }

</math>

</math>

数学

===''' 自由泊松分布Free Poisson distribution'''===

The free Poisson distribution<ref>Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992</ref> with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in [[free probability]] theory as the limit of repeated [[free convolution]]

The free Poisson distribution with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in free probability theory as the limit of repeated free convolution

带有跳跃大小 < math > alpha </math > 和速率 < math > lambda </math > 的''' 自由泊松分布Free Poisson distribution'''作为重复自由卷积的极限在自由概率论中出现

: <math>

<math>

《数学》

\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}</math>

\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}</math>

左(左(1-frac { lambda }{ n }右) delta _ 0 + frac { lambda }{ n } delta _ alpha 右) ^ { boxplus n } </math >

as ''N'' → ∞.

as N → ∞.

as N → ∞.

In other words, let <math>X_N</math> be random variables so that <math>X_N</math> has value <math>\alpha</math> with probability <math>\frac{\lambda}{N}</math> and value 0 with the remaining probability. Assume also that the family <math>X_1,X_2,\ldots</math> are [[free independence|freely independent]]. Then the limit as <math>N\to\infty</math> of the law of <math>X_1+\cdots +X_N</math>

In other words, let <math>X_N</math> be random variables so that <math>X_N</math> has value <math>\alpha</math> with probability <math>\frac{\lambda}{N}</math> and value 0 with the remaining probability. Assume also that the family <math>X_1,X_2,\ldots</math> are freely independent. Then the limit as <math>N\to\infty</math> of the law of <math>X_1+\cdots +X_N</math>

换句话说，让 x _ n </math > 是随机变量，因此 x _ n </math > 具有值 < math > alpha </math > 具有概率 < math > frac { lambda }{ n } </math > ，值0具有剩余的概率。同时假设家庭成员 x1，x2，ldots </math > 是自由独立的。然后将极限值设为 < math > n，以确定 < math > x1 + cdots + xn </math > 的规律
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“假设家庭成员 ”改为“假设集合 ”
is given by the Free Poisson law with parameters <math>\lambda,\alpha</math>.

is given by the Free Poisson law with parameters <math>\lambda,\alpha</math>.

是由带参数的自由泊松定律给出的。

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

这个定义类似于从(经典)泊松过程获得经典''' 泊松分佈Poisson distribution.'''的一种方法。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“泊松分布 ”

The measure associated to the free Poisson law is given by<ref>James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.</ref>

The measure associated to the free Poisson law is given by
与自由泊松定律相关的测度由以下给出

:<math>\mu=\begin{cases} (1-\lambda) \delta_0 + \lambda \nu,& \text{if } 0\leq \lambda \leq 1 \\

<math>\mu=\begin{cases} (1-\lambda) \delta_0 + \lambda \nu,& \text{if } 0\leq \lambda \leq 1 \\

< math > mu = begin { cases }(1-lambda) delta _ 0 + lambda nu，& text { if }0 leq lambda leq 1

\nu, & \text{if }\lambda >1,

\nu, & \text{if }\lambda >1,

1，& text { if } lambda > 1,

\end{cases}

\end{cases}

结束{ cases }

</math>

</math>

数学

where

where

其中

: <math>\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt</math>

<math>\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt</math>

4 lambda alpha ^ 2-(t-alpha (1 + lambda)) ^ 2} ，dt </math >

and has support <math>[\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2]</math>.

and has support <math>[\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2]</math>.

并且支持 < math > [ alpha (1-sqrt { lambda }) ^ 2，alpha (1 + sqrt { lambda }) ^ 2] </math > 。

This law also arises in [[random matrix]] theory as the [[Marchenko–Pastur law]]. Its [[free cumulants]] are equal to <math>\kappa_n=\lambda\alpha^n</math>.

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to <math>\kappa_n=\lambda\alpha^n</math>.

这个定律也出现在随机矩阵理论中，称为马尔琴科-帕斯图定律。它的自由累积量等于 < math > kappa _ n = lambda alpha ^ n </math > 。

====Some transforms of this law这一定律的一些变换====

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book ''Lectures on the Combinatorics of Free Probability'' by A. Nica and R. Speicher<ref>Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006</ref>

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher

我们给出了自由泊松定律的一些重要变换的值。在 a. Nica 和 r. Speicher 合著的《自由概率组合学讲座》一书中

The [[R-transform]] of the free Poisson law is given by

The R-transform of the free Poisson law is given by

以下给出自由泊松定律的 r- 变换

: <math>R(z)=\frac{\lambda \alpha}{1-\alpha z}. </math>

<math>R(z)=\frac{\lambda \alpha}{1-\alpha z}. </math>

1-alpha z }.数学

The [[Cauchy transform]] (which is the negative of the [[Stieltjes transformation]]) is given by

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by

柯西变换(即 Stieltjes 变换的负变换)由以下给出

: <math>

<math>

《数学》

G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}

G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}

G (z) = frac { z + alpha-lambda alpha-sqrt {(z-alpha (1 + lambda)))) ^ 2-4 lambda alpha ^ 2}{2 alpha z }

</math>

</math>

数学

The [[S-transform]] is given by

The S-transform is given by

以下给出了 s 变换的一般形式

: <math>

<math>

《数学》

S(z) = \frac{1}{z+\lambda}

S(z) = \frac{1}{z+\lambda}

(z) = frac {1}{ z + lambda }

</math>

</math>

数学

in the case that <math>\alpha=1</math>.

in the case that <math>\alpha=1</math>.

在这种情况下。

== Statistical Inference 统计学推论==

{{See also|Poisson regression}}
{{又见|泊松回归}}

=== Parameter estimation参数估计 ===

Given a sample of ''n'' measured values <math> k_i \in \{0,1,...\}</math>, for ''i'' = 1, ..., ''n'', we wish to estimate the value of the parameter ''λ'' of the Poisson population from which the sample was drawn. The [[maximum likelihood]] estimate is <ref>{{cite web |last=Paszek|first=Ewa |title=Maximum Likelihood Estimation – Examples |url = http://cnx.org/content/m13500/latest/?collection=col10343/latest}}</ref>

Given a sample of n measured values <math> k_i \in \{0,1,...\}</math>, for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is

给定一个 n 个测量值的样本{0,1，... } </math > ，对于 i = 1，... ，n，我们希望估计取样的泊松总体参数的值。''' 最大似然估计The maximum likelihood estimate'''是

: <math>\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i. \!</math>

<math>\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i. \!</math>

1}{ n } sum { i = 1} ^ n ki.! 数学

Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an [[unbiased estimator]] of λ. It is also an efficient estimator since its variance achieves the [[Cramér–Rao lower bound]] (CRLB).{{Citation needed|date=April 2012}} Hence it is [[Minimum-variance unbiased estimator|minimum-variance unbiased]]. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

因为每个观测值都有期望值，所以样本的意义也是如此。因此，''' 最大似然估计The maximum likelihood estimate'''是。由于其方差达到了 CRLB 下界，因此它也是一个有效的估计量。它是最小方差无偏的。也可以证明和(因此样本平均值是和的单射)是一个完整充分的统计量。

To prove sufficiency we may use the [[Sufficient statistic|factorization theorem]]. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math>\mathbf{x}</math> (called <math>h(\mathbf{x})</math>) and one that depends on the parameter <math>\lambda</math> and the sample <math>\mathbf{x}</math> only through the function <math>T(\mathbf{x})</math>. Then <math>T(\mathbf{x})</math> is a sufficient statistic for <math>\lambda</math>.

To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math>\mathbf{x}</math> (called <math>h(\mathbf{x})</math>) and one that depends on the parameter <math>\lambda</math> and the sample <math>\mathbf{x}</math> only through the function <math>T(\mathbf{x})</math>. Then <math>T(\mathbf{x})</math> is a sufficient statistic for <math>\lambda</math>.

为了证明充分性，我们可以用''' 因子分解定理Factorization theorem'''。考虑将''' 联合泊松分布Joint Poisson distribution'''的''' 概率质量函数Probability mass function'''分成两部分: 一部分仅依赖于样本 < math > mathbf { x } </math > (称为 < math > h (mathbf { x }) </math >) ，另一部分依赖于参数 < math > lambda </math > 和样本 < math > mathbf { x } </math > 只通过函数 math < t (mathbf { x }) </math > 。那么 < math > t (mathbf { x }) </math > 就是 < math > lambda </math > 的一个充分的统计量。

: <math> P(\mathbf{x})=\prod_{i=1}^n\frac{\lambda^{x_i} e^{-\lambda}}{x_i!}=\frac{1}{\prod_{i=1}^n x_i!} \times \lambda^{\sum_{i=1}^n x_i}e^{-n\lambda} </math>

<math> P(\mathbf{x})=\prod_{i=1}^n\frac{\lambda^{x_i} e^{-\lambda}}{x_i!}=\frac{1}{\prod_{i=1}^n x_i!} \times \lambda^{\sum_{i=1}^n x_i}e^{-n\lambda} </math>

= prod _ { i = 1} ^ n frac { lambda ^ { x _ i } e ^ {-lambda }{ x _ i！1}{ prod { i = 1} ^ n x i! }乘以 lambda ^ { sum { i = 1} ^ n xi } e ^ {-n lambda } </math >

The first term, <math>h(\mathbf{x})</math>, depends only on <math>\mathbf{x}</math>. The second term, <math>g(T(\mathbf{x})|\lambda)</math>, depends on the sample only through <math>T(\mathbf{x})=\sum_{i=1}^nx_i</math>. Thus, <math>T(\mathbf{x})</math> is sufficient.

The first term, <math>h(\mathbf{x})</math>, depends only on <math>\mathbf{x}</math>. The second term, <math>g(T(\mathbf{x})|\lambda)</math>, depends on the sample only through <math>T(\mathbf{x})=\sum_{i=1}^nx_i</math>. Thus, <math>T(\mathbf{x})</math> is sufficient.

第一个术语，< math > h (mathbf { x }) </math > ，仅依赖于 < math > mathbf { x } </math > 。第二个术语，< math > g (t (mathbf { x }) | lambda) </math > ，仅通过 < math > t (mathbf { x }) = sum { i = 1} ^ nx _ i </math > 取决于样本。因此，< math > t (mathbf { x }) </math > 就足够了。

To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:

To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:

为了找到泊松族群概率密度函数最大的参数λ ，我们可以使用''' 似然函数Likelihood function'''的对数:

: <math> \begin{align} \ell(\lambda) & = \ln \prod_{i=1}^n f(k_i \mid \lambda) \\ & = \sum_{i=1}^n \ln\!\left(\frac{e^{-\lambda}\lambda^{k_i}}{k_i!}\right) \\ & = -n\lambda + \left(\sum_{i=1}^n k_i\right) \ln(\lambda) - \sum_{i=1}^n \ln(k_i!). \end{align} </math>

<math> \begin{align} \ell(\lambda) & = \ln \prod_{i=1}^n f(k_i \mid \lambda) \\ & = \sum_{i=1}^n \ln\!\left(\frac{e^{-\lambda}\lambda^{k_i}}{k_i!}\right) \\ & = -n\lambda + \left(\sum_{i=1}^n k_i\right) \ln(\lambda) - \sum_{i=1}^n \ln(k_i!). \end{align} </math>

{ align } ell (lambda) & = ln prod { i = 1} ^ n f (k _ i mid lambda) & = sum { i = 1} ^ n ln! left (frac { e ^ {-lambda } lambda ^ { k _ i }{ k _ i！} right) & =-n lambda + left (sum _ { i = 1} ^ n k _ i right) ln (lambda)-sum _ { i = 1} ^ n ln (k _ i!).结束{ align } </math >

We take the derivative of ''<math>\ell</math>'' with respect to ''λ'' and compare it to zero:

We take the derivative of <math>\ell</math> with respect to λ and compare it to zero:

我们对 < math > 求导，然后将其与零进行比较:

: <math>\frac{\mathrm{d}}{\mathrm{d}\lambda} \ell(\lambda) = 0 \iff -n + \left(\sum_{i=1}^n k_i\right) \frac{1}{\lambda} = 0. \!</math>

<math>\frac{\mathrm{d}}{\mathrm{d}\lambda} \ell(\lambda) = 0 \iff -n + \left(\sum_{i=1}^n k_i\right) \frac{1}{\lambda} = 0. \!</math>

1}{ lambda } = 0 iff-n + left (sum { i = 1} ^ n k i right) frac {1}{ lambda } = 0.! 数学

Solving for ''λ'' gives a stationary point.

Solving for λ gives a stationary point.

解出 λ得到''' 驻点Stationary point'''。

: <math> \lambda = \frac{\sum_{i=1}^n k_i}{n}</math>

<math> \lambda = \frac{\sum_{i=1}^n k_i}{n}</math>

{ math > lambda = frac { sum { i = 1} ^ n k _ i }{ n } </math >

So ''λ'' is the average of the ''k''''i'' values. Obtaining the sign of the second derivative of ''L'' at the stationary point will determine what kind of extreme value ''λ'' is.

So λ is the average of the ki values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is.

K i 值的平均值也是如此。在驻点得到 l 的二阶导数的符号将决定什么是极值。

: <math>\frac{\partial^2 \ell}{\partial \lambda^2} = -\lambda^{-2}\sum_{i=1}^n k_i </math>

<math>\frac{\partial^2 \ell}{\partial \lambda^2} = -\lambda^{-2}\sum_{i=1}^n k_i </math>

{ partial ^ 2 ell }{ partial lambda ^ 2} =-lambda ^ {-2} sum { i = 1} ^ n k i </math >

Evaluating the second derivative ''at the stationary point'' gives:

Evaluating the second derivative at the stationary point gives:

在驻点对二阶导数进行评估得出:

: <math>\frac{\partial^2 \ell}{\partial \lambda^2} = - \frac{n^2}{\sum_{i=1}^n k_i} </math>

<math>\frac{\partial^2 \ell}{\partial \lambda^2} = - \frac{n^2}{\sum_{i=1}^n k_i} </math>

{ partial ^ 2 ell }{ partial lambda ^ 2} =-frac { n ^ 2}{ sum { i = 1} ^ n ki } </math >

which is the negative of ''n'' times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.

which is the negative of n times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.

它是 n 乘以 k i 平均值的倒数。当平均数为正时，这个表达式是负的。如果这一点得到了满足，那么''' 驻点The stationary point'''最大化了概率密度函数。

For [[Completeness (statistics)|completeness]], a family of distributions is said to be complete if and only if <math> E(g(T)) = 0</math> implies that <math>P_\lambda(g(T) = 0) = 1</math> for all <math>\lambda</math>. If the individual <math>X_i</math> are iid <math>\mathrm{Po}(\lambda)</math>, then <math>T(\mathbf{x})=\sum_{i=1}^nX_i\sim \mathrm{Po}(n\lambda)</math>. Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.

For completeness, a family of distributions is said to be complete if and only if <math> E(g(T)) = 0</math> implies that <math>P_\lambda(g(T) = 0) = 1</math> for all <math>\lambda</math>. If the individual <math>X_i</math> are iid <math>\mathrm{Po}(\lambda)</math>, then <math>T(\mathbf{x})=\sum_{i=1}^nX_i\sim \mathrm{Po}(n\lambda)</math>. Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.

为了完整性起见，一个分布族被认为是完整的，当且仅当 < math > e (g (t)) = 0 </math > 意味着 < math > p lambda (g (t) = 0) = 1 </math > 对于所有 < math > λ </math > 。如果个体 < math > xi </math > 是 < math > mathrm { Po }(λ) </math > ，那么 < math > t (mathbf { x }) = sum { i = 1} ^ nX _ i sim mathrm { Po }(n lambda) </math > 。了解了我们要调查的分布情况后，很容易看出统计数据是完整的。

: <math>E(g(T))=\sum_{t=0}^\infty g(t)\frac{(n\lambda)^te^{-n\lambda}}{t!}=0</math>

<math>E(g(T))=\sum_{t=0}^\infty g(t)\frac{(n\lambda)^te^{-n\lambda}}{t!}=0</math>

< math > e (g (t)) = sum { t = 0} ^ infty g (t) frac {(n lambda) ^ te ^ {-n lambda }{ t！0 </math >

For this equality to hold, <math>g(t)</math> must be 0. This follows from the fact that none of the other terms will be 0 for all <math>t</math> in the sum and for all possible values of <math>\lambda</math>. Hence, <math> E(g(T)) = 0</math> for all <math>\lambda</math> implies that <math>P_\lambda(g(T) = 0) = 1</math>, and the statistic has been shown to be complete.

For this equality to hold, <math>g(t)</math> must be 0. This follows from the fact that none of the other terms will be 0 for all <math>t</math> in the sum and for all possible values of <math>\lambda</math>. Hence, <math> E(g(T)) = 0</math> for all <math>\lambda</math> implies that <math>P_\lambda(g(T) = 0) = 1</math>, and the statistic has been shown to be complete.

要保证这个等式成立，< math > g (t) </math > 必须为0。这源于这样一个事实: 对于所有 < math > t </math > 的和和以及 < math > > lambda </math > 的所有可能值，其他项都不会为0。因此，e (g (t)) = 0 </math > > lambda </math > 意味着 < math > p _ lambda (g (t) = 0 = 1 </math > ，统计已被证明是完整的。

=== Confidence interval '''置信区间Confidence interval '''===

The [[confidence interval]] for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and [[chi-squared distribution]]s. The chi-squared distribution is itself closely related to the [[gamma distribution]], and this leads to an alternative expression. Given an observation ''k'' from a Poisson distribution with mean ''μ'', a confidence interval for ''μ'' with confidence level {{math|1 – α}} is

The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is

'''置信区间Confidence interval '''的平均''' 泊松分布Poisson distribution'''可以用泊松分布和卡方分布的累积分布函数之间的关系来表示。卡方分布本身与伽玛分布密切相关，这导致了另一种表达方式。给定一个来自平均泊松分佈的观测值 k，一个带有置信水平的置信区间是

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“泊松分布 ”

:<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>

<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>

12 chi ^ {2}(alpha/2; 2k) le mu le tfrac 12 chi ^ {2}(1-alpha/2; 2k + 2) ，</math >

or equivalently,

or equivalently,

或者等价,

:<math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math>

<math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math>

(1-alpha/2; k，1) le mu le f ^ {-1}(1-alpha/2; k + 1,1) ，</math >

where <math>\chi^{2}(p;n)</math> is the [[quantile function]] (corresponding to a lower tail area ''p'') of the chi-squared distribution with ''n'' degrees of freedom and <math>F^{-1}(p;n,1)</math> is the quantile function of a [[gamma distribution]] with shape parameter n and scale parameter 1.{{r|Johnson2005|p=176-178|Garwood1936}} This interval is '[[exact statistics|exact]]' in the sense that its [[coverage probability]] is never less than the nominal {{math|1 – α}}.

where <math>\chi^{2}(p;n)</math> is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and <math>F^{-1}(p;n,1)</math> is the quantile function of a gamma distribution with shape parameter n and scale parameter 1. This interval is 'exact' in the sense that its coverage probability is never less than the nominal .

其中 < math > chi ^ {2}(p; n) </math > 是 n 个自由度的分位函数，< math > f ^ {-1}(p; n，1) </math > 是形状参数 n 和尺度参数1的卡方分布的分位函数。这个时间间隔是“精确的” ，因为它的覆盖概率从来没有小于名义值。

When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the [[Chi-square distribution#Asymptotic properties|Wilson–Hilferty transformation]]):{{r|Breslow1987}}

When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):

当伽玛分布的分位数不可用时，对这个精确区间提出了精确的近似(基于 Wilson-Hilferty 变换) :

:<math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math>

<math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math>

< math > k left (1-frac {1}{9k }-frac { z _ { alpha/2}{3 sqrt { k }右) ^ 3 le mu le (k + 1) left (1-frac {1}{9(k + 1)}} + frac { z _ { alpha/2}{3 sqrt { k + 1}右) ^ 3，</math >

where <math>z_{\alpha/2}</math> denotes the [[standard normal deviate]] with upper tail area {{math|α / 2}}.

where <math>z_{\alpha/2}</math> denotes the standard normal deviate with upper tail area .

其中 < math > z _ { alpha/2} </math > 表示标准的正常偏差和上尾区。

For application of these formulae in the same context as above (given a sample of ''n'' measured values ''k''''i'' each drawn from a Poisson distribution with mean ''λ''), one would set

For application of these formulae in the same context as above (given a sample of n measured values ki each drawn from a Poisson distribution with mean λ), one would set

为了在与上述相同的上下文中应用这些公式(给定一个 n 个测量值 k i 每个取自一个泊松分佈的平均值) ，我们将设置

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“泊松分布 ”

:<math>k=\sum_{i=1}^n k_i ,\!</math>

<math>k=\sum_{i=1}^n k_i ,\!</math>

[ math > k = sum { i = 1} ^ n k _ i，

calculate an interval for ''μ'' = ''nλ'', and then derive the interval for ''λ''.

calculate an interval for μ = nλ, and then derive the interval for λ.

计算 = n 的区间，然后推导出。

=== ''' Bayesian inference 贝叶斯推理'''===

In [[Bayesian inference]], the [[conjugate prior]] for the rate parameter ''λ'' of the Poisson distribution is the [[gamma distribution]].{{r|Fink1976}} Let

In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. Let

在''' Bayesian inference 贝叶斯推理'''中，泊松分佈的速率参数的''' 共轭先验Conjugate prior'''是伽玛分布。让

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“泊松分布 ”

:<math>\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!</math>

<math>\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!</math>

{ Gamma }(alpha，beta)

denote that ''λ'' is distributed according to the gamma [[probability density function|density]] ''g'' parameterized in terms of a [[shape parameter]] ''α'' and an inverse [[scale parameter]] ''β'':

denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:

表示根据以形状参数和反比例尺参数表示的伽马密度 g 分布:

:<math> g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.</math>

<math> g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.</math>

G (lambda mid alpha，beta) = frac { beta ^ { alpha }{ Gamma (alpha)} ; lambda ^ { alpha-1} ; e ^ {-beta，lambda } qquad text { for } lambda > 0，! </math >

Then, given the same sample of ''n'' measured values ''k''''i'' [[Poisson distribution#Maximum likelihood|as before]], and a prior of Gamma(''α'', ''β''), the posterior distribution is

Then, given the same sample of n measured values ki as before, and a prior of Gamma(α, β), the posterior distribution is

然后，给定相同的样本 n 测量值 k i 和之前的 Gamma (,) ，后验概率是

:<math>\lambda \sim \mathrm{Gamma}\left(\alpha + \sum_{i=1}^n k_i, \beta + n\right). \!</math>

<math>\lambda \sim \mathrm{Gamma}\left(\alpha + \sum_{i=1}^n k_i, \beta + n\right). \!</math>

左(alpha + sum _ { i = 1} ^ n k _ i，beta + n right)。! 数学

The posterior mean E[''λ''] approaches the maximum likelihood estimate <math>\widehat{\lambda}_\mathrm{MLE}</math> in the limit as <math>\alpha\to 0,\ \beta\to 0</math>, which follows immediately from the general expression of the mean of the [[gamma distribution]].

The posterior mean E[λ] approaches the maximum likelihood estimate <math>\widehat{\lambda}_\mathrm{MLE}</math> in the limit as <math>\alpha\to 0,\ \beta\to 0</math>, which follows immediately from the general expression of the mean of the gamma distribution.

后验平均值 e []接近极限中的最大似然估计 < math > lambda } _ mathrm { MLE } </math > ，它紧跟在伽玛分布平均值的一般表达式之后。

The [[posterior predictive distribution]] for a single additional observation is a [[negative binomial distribution]],{{r|Gelman2003|p=53}} sometimes called a gamma–Poisson distribution.

The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a gamma–Poisson distribution.

单一额外观察的后验预测分布是负二项分布，有时称为泊松分佈。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“泊松分布 ”

=== Simultaneous estimation of multiple Poisson means 多重泊松均值的同步估计===

Suppose <math>X_1, X_2, \dots, X_p</math> is a set of independent random variables from a set of <math>p</math> Poisson distributions, each with a parameter <math>\lambda_i</math>, <math>i=1,\dots,p</math>, and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss <math>L(\lambda,{\hat \lambda})=\sum_{i=1}^p \lambda_i^{-1} ({\hat \lambda}_i-\lambda_i)^2</math>, when <math>p>1</math>, then, similar as in [[Stein's example]] for the Normal means, the MLE estimator <math>{\hat \lambda}_i = X_i</math> is [[Admissible decision rule|inadmissible]]. {{r|Clevenson1975}}

Suppose <math>X_1, X_2, \dots, X_p</math> is a set of independent random variables from a set of <math>p</math> Poisson distributions, each with a parameter <math>\lambda_i</math>, <math>i=1,\dots,p</math>, and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss <math>L(\lambda,{\hat \lambda})=\sum_{i=1}^p \lambda_i^{-1} ({\hat \lambda}_i-\lambda_i)^2</math>, when <math>p>1</math>, then, similar as in Stein's example for the Normal means, the MLE estimator <math>{\hat \lambda}_i = X_i</math> is inadmissible.

假设 x _ 1，x _ 2，点，x _ p </math > 是一组来自一组 p </math > 泊松分布的独立随机变量，每个分布都有一个参数 λ < math > i </math > ，< math > i = 1，点，p </math > ，我们想估计这些参数。然后，clevensen 和 Zidek 证明，在归一化的平方误差损失 < math > l (lambda，{ hat lambda }) = sum { i = 1} ^ p lambda i ^ {-1}({ hat lambda } i-lambda _ i) ^ 2 </math > 下，当 < math > p > 1 </math > ，那么，类似于 Stein 的例子中的正态方法，MLE < math > lambda } i = xi </math > 是不允许的。

In this case, a family of [[minimax estimator]]s is given for any <math>0 < c \leq 2(p-1)</math> and <math>b \geq (p-2+p^{-1})</math> as{{r|Berger1985}}

In this case, a family of minimax estimators is given for any <math>0 < c \leq 2(p-1)</math> and <math>b \geq (p-2+p^{-1})</math> as

在这种情况下，对于任意 < math > 0 < c leq 2(p-1) </math > 和 < math > b geq (p-2 + p ^ {-1}) </math > ，给出了极大极小估计族

:<math>{\hat \lambda}_i = \left(1 - \frac{c}{b + \sum_{i=1}^p X_i}\right) X_i, \qquad i=1,\dots,p.</math>

<math>{\hat \lambda}_i = \left(1 - \frac{c}{b + \sum_{i=1}^p X_i}\right) X_i, \qquad i=1,\dots,p.</math>

{ hat lambda } _ i = left (1-frac { c }{ b + sum { i = 1} ^ p x _ i } right) xi，qquad i = 1，dots，p </math >

== Occurrence and applications 发生与应用==

{{More citations needed|date=December 2019}}

Applications of the Poisson distribution can be found in many fields including:{{r|Rasch1963}}

Applications of the Poisson distribution can be found in many fields including:

''' 泊松分佈Poisson distribution'''可以应用在很多领域，包括:
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”
* [[Telecommunication]] example: telephone calls arriving in a system.
*[[电信]]示例：到达系统的电话。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“到达系统的电话”改为“一个系统中的来电数 ”
* [[Astronomy]] example: photons arriving at a telescope.
*[[天文学]]例子：到达望远镜的光子。
* [[Chemistry]] example: the [[molar mass distribution]] of a [[living polymerization]].{{r|Flory1940}}
*[[化学]]示例：一种[[活性聚合物]的[[摩尔质量分布]]。
* [[Biology]] example: the number of mutations on a strand of [[DNA]] per unit length.
*[[生物]]示例：每单位[[DNA]]链上的突变数。
* [[Management]] example: customers arriving at a counter or call centre.
*[[管理]]示例：到达柜台或呼叫中心的客户。
* [[Finance and insurance]] example: number of losses or claims occurring in a given period of time.
*[[金融和保险]]示例：在一定时期内发生的损失或索赔的数量。
* [[Earthquake seismology]] example: an asymptotic Poisson model of seismic risk for large earthquakes.{{r|Lomnitz1994|p=}}
*[[地震地震学]]实例：大地震地震危险性的渐近泊松模型。{{r|Lomnitz1994|p=}}
* [[Radioactivity]] example: number of decays in a given time interval in a radioactive sample.
*[[放射性]]示例：在放射性样本中给定时间间隔内的衰变次数。
* [[Optics]] example: the number of photons emitted in a single laser pulse. This is a major vulnerability to most [[Quantum key distribution]] protocols known as Photon Number Splitting (PNS).
*[[光学]]示例：单个激光脉冲中发射的光子数。这是大多数[[量子密钥分发]]协议的主要漏洞，被称为光子数分裂（PNS）。

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or [[space]]. Examples of events that may be modelled as a Poisson distribution include:

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:

泊松分佈过程与泊松过程有关。它适用于各种离散性质的现象(也就是说，那些可能发生0,1,2,3，... 在给定时间内或在给定区域) ，只要现象发生的概率在时间或空间上是常数。可以被模仿为泊松分佈的活动包括:
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”



-->

-->

* The number of soldiers killed by horse-kicks each year in each corps in the [[Prussia]]n cavalry. This example was used in a book by [[Ladislaus Bortkiewicz]] (1868–1931).{{r|vonBortkiewitsch1898|p=23-25}}
*各兵团每年死于马踢的士兵人数。这个例子在[[Ladislaus Bortkiewicz]的一本书中使用过(1868–1931).{{r|vonBortkiewitsch1898|p=23-25}}。
* The number of yeast cells used when brewing [[Guinness]] beer. This example was used by [[William Sealy Gosset]] (1876–1937).{{r|Student1907}}{{r|Boland1984}}
*酿造[[吉尼斯]]啤酒时使用的酵母细胞数量。这个例子被[[William sely Gosset]]（1876-1937）使用。{{r|Student1907}}{{r|Boland1984}}
* The number of phone calls arriving at a [[call centre]] within a minute. This example was described by [[Agner Krarup Erlang|A.K. Erlang]] (1878–1929).{{r|Erlang1909}}
*一分钟内到达[[呼叫中心]]的电话数。这个例子由[[Agner Krarup Erlang | A.K.Erlang]]（1878-1929）描述{{r|Erlang1909}}。
* Internet traffic.
*互联网堵塞。
* The number of goals in sports involving two competing teams.{{r|Hornby2014}}
*两支参赛队伍在运动中的进球数。{{r|Hornby2014}}
* The number of deaths per year in a given age group.
*特定年龄组每年的死亡人数。
* The number of jumps in a stock price in a given time interval.
*股票价格在给定时间间隔内的波动次数。
* Under an assumption of [[Poisson process#Homogeneous|homogeneity]], the number of times a [[web server]] is accessed per minute.
*在[[泊松过程#齐次|均匀性]假设下，每分钟访问[[web服务器]]的次数。
* The number of [[mutation]]s in a given stretch of [[DNA]] after a certain amount of radiation.
*在一定量的辐射之后，在给定的[[DNA]]段中[[突变]]的数目。
* The proportion of [[cells (biology)|cells]] that will be infected at a given [[multiplicity of infection]].
*在给定的时间内被感染的[[细胞（生物学）|细胞]]的比例。
* The number of bacteria in a certain amount of liquid.{{r|Koyama2016}}
*一定量液体中细菌的数量。{{r|Koyama2016}}
* The arrival of [[photons]] on a pixel circuit at a given illumination and over a given time period.
*在给定的光照和时间间隔，到达像素电路的[[光子]]。
* The targeting of [[V-1 flying bomb]]s on London during World War II investigated by R. D. Clarke in 1946.{{r|Clarke1946}}
*二战期间伦敦对[[V-1飞弹]]的目标调查||由R. D. Clarke 1946年调查。{{r|Clarke1946}}
[[Patrick X. Gallagher|Gallagher]] showed in 1976 that the counts of [[prime number]]s in short intervals obey a Poisson distribution{{r|Gallagher1976}} provided a certain version of the unproved [[Second Hardy–Littlewood conjecture|prime r-tuple conjecture of Hardy-Littlewood]]{{r|Hardy1923}} is true.

Gallagher showed in 1976 that the counts of prime numbers in short intervals obey a Poisson distribution provided a certain version of the unproved prime r-tuple conjecture of Hardy-Littlewood is true.

1976年，加拉格尔指出，只要Hardy-Littlewo素数r-元组猜想的一个版本为正确，短时间间隔内质数的计数即服从泊松分布。

{{Anchor|law of rare events}}

=== Law of rare events稀有事件定律 ===

{{main|Poisson limit theorem}}

[[File:Binomial versus poisson.svg|right|upright=1.5|thumb |Comparison of the Poisson distribution (black lines) and the [[binomial distribution]] with ''n'' = 10 (red circles), ''n'' = 20 (blue circles), ''n'' = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events ''k''. As ''n'' gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.]]

Comparison of the Poisson distribution (black lines) and the [[binomial distribution with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events k. As n gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.]]

泊松分佈(黑线)与[二项分布(红圈) ，n = 20(蓝圈) ，n = 1000(绿圈)的比较。所有分布的平均值都是5。水平轴显示事件的数量 k。随着 n 变得越来越大，泊松分佈变成了一个越来越好的平均二项分布。]
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a [[Bernoulli trial]], where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.

The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a Bernoulli trial, where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.

事件的发生率与事件发生在某个小的子间隔(时间、空间或其他)的概率有关。在泊松分佈的例子中，我们假设存在一个足够小的子区间，其中一个事件发生两次的概率是“可以忽略的”。有了这个假设，我们就可以从二项式中推导出泊松分佈，只需要给出整个时间间隔内预期的事件总数的信息。设这个总数是 < math > > lambda </math > 。将整个区间分为 < math > n </math > 子区间 < math > i _ 1，点，i _ n </math > 大小相等，这样 < math > n </math > < math > lambda </math > (因为我们只对区间的很小一部分感兴趣，所以这个假设是有意义的)。这意味着每个 < math > i </math > 中期望的事件数等于 < math > lambda/n </math > 。现在，我们假设一个事件在整个时间间隔内的发生可以被看作是伯努利试验，其中，“ math”试验对应于观察一个事件是否在子时间间隔内发生。在 < math > n </math > 这样的试验中预期的总事件数是 < math > lambda </math > ，这是整个间隔中预期的总事件数。因此，对于区间的每一个细分，我们都近似地将事件的发生作为形式 < math > textrm { b }(n，lambda/n) </math > 的伯努利过程。正如我们之前指出的，我们只想考虑非常小的子区间。因此，我们将极限取为 < math > n </math > 到无穷大。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”
In this case the binomial distribution converges to what is known as the Poisson distribution by the [[Poisson limit theorem]].

In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.

在这种情况下，二项分布收敛于泊松极限定理所称的泊松分佈。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the [[binomial distribution]], that is

In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is

在上面的几个例子中---- 例如，一个给定序列的 dna 突变的数量---- 被计算的事件实际上是离散试验的结果，也就是说，更准确地说，是用二项分布模型来模拟的

:<math>X \sim \textrm{B}(n,p). \,</math>

<math>X \sim \textrm{B}(n,p). \,</math>

X sim textrm { b }(n，p).，math

In such cases ''n'' is very large and ''p'' is very small (and so the expectation ''np'' is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution{{Citation needed|date=April 2012}}

In such cases n is very large and p is very small (and so the expectation np is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution

在这种情况下，n 是非常大的，p 是非常小的(所以期望 np 是中等大小)。然后，分布可以近似于不那么麻烦的泊松分佈

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

:<math>X \sim \textrm{Pois}(np). \,</math>

<math>X \sim \textrm{Pois}(np). \,</math>

<math>X \sim \textrm{Pois}(np).，math

This approximation is sometimes known as the ''law of rare events'',{{r|Cameron1998|p=5}}since each of the ''n'' individual [[Bernoulli distribution|Bernoulli events]] rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter ''np'' is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

This approximation is sometimes known as the law of rare events,since each of the n individual Bernoulli events rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter np is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

这种近似有时被称为稀有事件定律，因为 n 个伯努利事件中的每一个很少发生。这个名称可能有误导性，因为如果参数 np 不小，那么 Poisson 过程中成功事件的总计数就不会很少。例如，一个小时内打给忙碌总机的电话数量跟随着一个泊松分佈，这些事件在接线员看来是频繁的，但是从普通人的角度来看，这些事件很少发生，因为他们不太可能在那个小时内打电话给总机。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

The word ''law'' is sometimes used as a synonym of [[probability distribution]], and ''convergence in law'' means ''convergence in distribution''. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. ''The Law of Small Numbers'' is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.{{r|vonBortkiewitsch1898}}{{r|Edgeworth1913}}

The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.

法律一词有时被用作概率分布的同义词，法律的趋同意味着分配的趋同。因此，泊松分佈有时被称为“小数定律” ，因为它是一个事件发生次数的概率分布，这个事件很少发生，但却有很多机会发生。小数定律》是拉迪斯劳斯·博特基威茨的一本关于泊松分佈的书，出版于1898年。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

===''' Poisson point process 泊松点过程'''===

{{Main|Poisson point process}}

The Poisson distribution arises as the number of points of a [[Poisson point process]] located in some finite region. More specifically, if ''D'' is some region space, for example Euclidean space '''R'''''d'', for which |''D''|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if {{nowrap|''N''(''D'')}} denotes the number of points in ''D'', then

The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if denotes the number of points in D, then

泊松分佈是位于某个有限区域的泊松过程的点数。更具体地说，如果 d 是某个区域空间，例如欧几里德空间 r d ，对于这个区域 | d | ，区域的面积、体积或者更一般地说，区域的勒贝格测度是有限的，如果表示 d 中的点数，那么
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

: <math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>

<math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>

< math > p (n (d) = k) = frac {(lambda | d |) ^ k e ^ {-lambda | d | }{ k! }. math

=== ''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''===

[[Poisson regression]] and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.

Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.

''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''分析是有用的，其中依赖(响应)变量是计数(0,1,2，...)的在一个区间内事件发生的数量。

=== Other applications in science 科学上的其他应用===

In a Poisson process, the number of observed occurrences fluctuates about its mean ''λ'' with a [[standard deviation]] <math>\sigma_k =\sqrt{\lambda}</math>. These fluctuations are denoted as ''Poisson noise'' or (particularly in electronics) as ''[[shot noise]]''.

In a Poisson process, the number of observed occurrences fluctuates about its mean λ with a standard deviation <math>\sigma_k =\sqrt{\lambda}</math>. These fluctuations are denoted as Poisson noise or (particularly in electronics) as shot noise.

在泊松过程中，观察到的事件数目在其平均值上下波动，波动标准差为1/2。这些波动被称为泊松噪声或(特别是在电子学中)散粒噪声。

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, ''even if that contribution is too small to be detected directly''. For example, the charge ''e'' on an electron can be estimated by correlating the magnitude of an [[electric current]] with its [[shot noise]]. If ''N'' electrons pass a point in a given time ''t'' on the average, the [[mean]] [[Electric current|current]] is <math>I=eN/t</math>; since the current fluctuations should be of the order <math>\sigma_I=e\sqrt{N}/t</math> (i.e., the standard deviation of the [[Poisson process]]), the charge <math>e</math> can be estimated from the ratio <math>t\sigma_I^2/I</math>.{{Citation needed|date=April 2012}}

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If N electrons pass a point in a given time t on the average, the mean current is <math>I=eN/t</math>; since the current fluctuations should be of the order <math>\sigma_I=e\sqrt{N}/t</math> (i.e., the standard deviation of the Poisson process), the charge <math>e</math> can be estimated from the ratio <math>t\sigma_I^2/I</math>.

在计算独立的离散事件时，平均数和标准差的相关性是有科学价值的。通过监测波动是如何随着平均信号而变化的，我们可以估计单一事件的贡献，即使这个贡献太小而不能直接检测到。例如，电子的电荷 e 可以通过将电流的大小与散粒噪声相关联来估计。如果 n 个电子在给定时间 t 平均通过一个点，那么平均电流为 < math > i = eN/t </math > ; 因为当前的波动应该是 < math > sigma i = e sqrt { n }/t </math > (即 Poisson 过程的标准差) ，所以电荷 < math > e </math > 可以通过数学比率 < t sigma _ i ^ 2/I </math > 来估计。

An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced [[silver]] grains, not to the individual grains themselves. By [[Correlation|correlating]] the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).{{Citation needed|date=April 2012}} Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of [[receptor (biochemistry)|receptor]] molecules in a [[cell membrane]].

An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided). Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of receptor molecules in a cell membrane.

一个日常的例子是放大照片时出现的颗粒状; 颗粒状是由于减少的银粒数量的泊松波动，而不是单个颗粒本身。通过将颗粒度与放大程度相关联，我们可以估算出单个颗粒的贡献(否则颗粒太小，无法单独看到)。泊松噪声的许多其他分子应用已经发展起来，例如，估计细胞膜上受体分子的数量密度。

: <math>

<math>

《数学》

\Pr(N_t=k) = f(k;\lambda t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}.</math>

\Pr(N_t=k) = f(k;\lambda t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}.</math>

Pr (n _ t = k) = f (k; lambda t) = frac {((lambda t) ^ k e ^ {-lambda t }}{ k！} . </math >

In [[Causal Set]] theory the discrete elements of spacetime follow a Poisson distribution in the volume.

In Causal Set theory the discrete elements of spacetime follow a Poisson distribution in the volume.

在''' 因果集合论Causal Set theory'''中，时空的离散元素在集合中遵循一个泊松分佈。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

==Computational methods计算方法==

The Poisson distribution poses two different tasks for dedicated software libraries: ''Evaluating'' the distribution <math>P(k;\lambda)</math>, and ''drawing random numbers'' according to that distribution.

The Poisson distribution poses two different tasks for dedicated software libraries: Evaluating the distribution <math>P(k;\lambda)</math>, and drawing random numbers according to that distribution.

泊松分佈为专用软件库提出了两个不同的任务: 评估分布 < math > p (k; lambda) </math > ，并根据分布绘制随机数。

=== ''' Evaluating the Poisson distribution 计算泊松分布'''===

Computing <math>P(k;\lambda)</math> for given <math>k</math> and <math>\lambda</math> is a trivial task that can be accomplished by using the standard definition of <math>P(k;\lambda)</math> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λ''k'' and ''k''!. The fraction of λ''k'' to ''k''! can also produce a rounding error that is very large compared to ''e''−λ, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as

Computing <math>P(k;\lambda)</math> for given <math>k</math> and <math>\lambda</math> is a trivial task that can be accomplished by using the standard definition of <math>P(k;\lambda)</math> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λk and k!. The fraction of λk to k! can also produce a rounding error that is very large compared to e−λ, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as

计算 < math > p (k; lambda) </math > 对于给定的 < math > k </math > 和 < math > lambda </math > 是一项琐碎的任务，可以通过使用 < math > p (k; lambda) </math > 的标准定义来完成，包括指数函数、幂函数和阶乘函数。然而，传统上对泊松分佈的定义包含了两个容易在计算机上溢出的术语: k 和 k。分数 k 到 k！也可能产生舍入误差，与 e - 相比，舍入误差非常大，因此给出错误的结果。因此，对于数值稳定性来说，泊松概率质量函数应该被评估为
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

:<math>\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],</math>

<math>\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],</math>

= exp left [ k ln lambda-lambda-ln Gamma (k + 1) right ] ，</math >

which is mathematically equivalent but numerically stable. The natural logarithm of the [[Gamma function]] can be obtained using the <code>lgamma</code> function in the [[C (programming language)|C]] standard library (C99 version) or [[R (programming language)|R]], the <code>gammaln</code> function in [[MATLAB]] or [[SciPy]], or the <code>log_gamma</code> function in [[Fortran]] 2008 and later.

which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the <code>lgamma</code> function in the C standard library (C99 version) or R, the <code>gammaln</code> function in MATLAB or SciPy, or the <code>log_gamma</code> function in Fortran 2008 and later.

这在数学上是等价的，但在数值上是稳定的。函数的自然对数可以使用 c 标准库(C99版本)中的 < code > lgamma </code > 函数或 r，MATLAB 或 SciPy 中的 < code > gammaln </code > 函数，或 Fortran 2008及更高版本中的 < code > log _ Gamma </code > 函数来获得。

Some computing languages provide built-in functions to evaluate the Poisson distribution, namely

Some computing languages provide built-in functions to evaluate the Poisson distribution, namely

一些计算语言提供了''' 内置函数Built-in functions'''来评估泊松分佈
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”
* [[R (programming language)|R]]: function <code>dpois(x, lambda)</code>;

* [[Microsoft Excel|Excel]]: function <code>POISSON( x, mean, cumulative)</code>, with a flag to specify the cumulative distribution;

* [[Mathematica]]: univariate Poisson distribution as <code>PoissonDistribution[<math>\lambda</math>]</code>,<ref name="WLPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/PoissonDistribution.html |title = Wolfram Language: PoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref> bivariate Poisson distribution as <code>MultivariatePoissonDistribution[<math>\theta_{12}</math>,{ <math>\theta_1 - \theta_{12}</math>, <math>\theta_2 - \theta_{12}</math>}]</code>,.<ref name="WLMvPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html |title = Wolfram Language: MultivariatePoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref>

=== Random drawing from the Poisson distribution 从泊松分布中抽取随机量===

The less trivial task is to draw random integers from the Poisson distribution with given <math>\lambda</math>.

The less trivial task is to draw random integers from the Poisson distribution with given <math>\lambda</math>.

更简单的任务是用给定的 < math > lambda </math > 从泊松分佈中提取随机整数。

Solutions are provided by:

Solutions are provided by:

提供解决方案的有:

* [[R (programming language)|R]]: function <code>rpois(n, lambda)</code>;

* [[GNU Scientific Library]] (GSL): function [https://www.gnu.org/software/gsl/doc/html/randist.html#the-poisson-distribution gsl_ran_poisson]

=== Generating Poisson-distributed random variables 生成泊松分布随机变量===

A simple algorithm to generate random Poisson-distributed numbers ([[pseudo-random number sampling]]) has been given by [[Donald Knuth|Knuth]]:{{r|Knuth1997|p=137-138}}

A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth:

给出了一个产生随机泊松分布数(伪随机数抽样)的简单算法:

'''algorithm''' ''poisson random number (Knuth)'':

algorithm poisson random number (Knuth):

算法泊松随机数(Knuth) :

'''init''':

init:

初始化:

'''Let''' L ← ''e''−λ, k ← 0 and p ← 1.

Let L ← e−λ, k ← 0 and p ← 1.

Let L ← e−λ, k ← 0 and p ← 1.

'''do''':

do:

做:

k ← k + 1.

k ← k + 1.

k ← k + 1.

Generate uniform random number u in [0,1] and '''let''' p ← p × u.

Generate uniform random number u in [0,1] and let p ← p × u.

在[0,1]中生成均匀随机数 u 并且设 p ← p u。

'''while''' p > L.

while p > L.

而 p > l。

'''return''' k − 1.

return k − 1.

return k − 1.

The complexity is linear in the returned value ''k'', which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see {{slink||References}} below.

The complexity is linear in the returned value k, which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see below.

返回值 k 的复杂度是线性的，平均为。还有许多其他的算法可以改进这一点。一些是在 Ahrens & Dieter，见下面。

For large values of λ, the value of L = ''e''−λ may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that ''e''−STEP does not underflow: {{Citation needed|reason=Original source is missing|date=March 2019}}

For large values of λ, the value of L = e−λ may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e−STEP does not underflow:

对于较大的值，l = e - 的值可能非常小，以至于很难表示。这可以通过改变算法来解决，该算法使用附加参数 STEP，使得 e -STEP 不会底流:

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“底流”改为“下溢”

'''algorithm''' ''poisson random number (Junhao, based on Knuth)'':

algorithm poisson random number (Junhao, based on Knuth):

泊松随机数算法(Junhao，基于 Knuth) :

'''init''':

init:

初始化:

'''Let''' λLeft ← λ, k ← 0 and p ← 1.

Let λLeft ← λ, k ← 0 and p ← 1.

Let λLeft ← λ, k ← 0 and p ← 1.

'''do''':

do:

做:

k ← k + 1.

k ← k + 1.

k ← k + 1.

Generate uniform random number u in (0,1) and '''let''' p ← p × u.

Generate uniform random number u in (0,1) and let p ← p × u.

在(0,1)中生成均匀随机数 u 并且设 p ← p u。

'''while''' p < 1 and λLeft > 0:

while p < 1 and λLeft > 0:

while p < 1 and λLeft > 0:

'''if''' λLeft > STEP:

if λLeft > STEP:

if λLeft > STEP:

p ← p × ''e''STEP

p ← p × eSTEP

p ← p × eSTEP

λLeft ← λLeft − STEP

λLeft ← λLeft − STEP

λLeft ← λLeft − STEP

'''else''':

else:

其他:

p ← p × ''e''λLeft

p ← p × eλLeft

p ← p × eλLeft

λLeft ← 0

λLeft ← 0

λLeft ← 0

'''while''' p > 1.

while p > 1.

同时 p > 1。

'''return''' k − 1.

return k − 1.

return k − 1.

The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near ''e''700, so 500 shall be a safe ''STEP''.

The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near e700, so 500 shall be a safe STEP.

STEP 的选择取决于溢出''' 阈值Threshold'''。对于双精度浮点格式，''' 阈值Threshold'''接近 e 700 ，因此500应该是一个安全的 STEP。

Other solutions for large values of λ include [[rejection sampling]] and using Gaussian approximation.

Other solutions for large values of λ include rejection sampling and using Gaussian approximation.

其他λ的大数值的求解包括抑制取样和使用''' 高斯近似Gaussian approximation'''。

[[Inverse transform sampling]] is simple and efficient for small values of λ, and requires only one uniform random number ''u'' per sample. Cumulative probabilities are examined in turn until one exceeds ''u''.

Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.

逆变换采样对于λ小数值的样本是简单有效的，并且每个样本只需要一个均匀一致的随机数 u。依次检查累积概率，直到其超过 u。

'''algorithm''' ''Poisson generator based upon the inversion by sequential search'':{{r|Devroye1986|p=505}}

algorithm Poisson generator based upon the inversion by sequential search:

基于顺序检索反演的''' 泊松发生器算法Algorithm Poisson generator ''':

'''init''':

init:

初始化:

'''Let''' x ← 0, p ← ''e''−λ, s ← p.

Let x ← 0, p ← e−λ, s ← p.

Let x ← 0, p ← e−λ, s ← p.

Generate uniform random number u in [0,1].

Generate uniform random number u in [0,1].

在[0,1]中生成均匀随机数 u。

'''while''' u > s '''do''':

while u > s do:

做以下步骤:

x ← x + 1.

x ← x + 1.

x ← x + 1.

p ← p × λ / x.

p ← p × λ / x.

p ← p × λ / x.

s ← s + p.

s ← s + p.

s ← s + p.

'''return''' x.

return x.

返回 x。

== History历史 ==

The distribution was first introduced by [[Siméon Denis Poisson]] (1781–1840) and published together with his probability theory in his work ''Recherches sur la probabilité des jugements en matière criminelle et en matière civile''(1837).{{r|Poisson1837|p=205-207}} The work theorized about the number of wrongful convictions in a given country by focusing on certain [[random variable]]s ''N'' that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a [[time]]-interval of given length. The result had already been given in 1711 by [[Abraham de Moivre]] in ''De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus'' .{{r|deMoivre1711|p=219}}{{r|deMoivre1718|p=14-15}}{{r|deMoivre1721|p=193}}{{r|Johnson2005|p=157}} This makes it an example of [[Stigler's law]] and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.{{r|Stigler1982|Hald1984}}

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile(1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile(1837).
这种分布最早由西蒙·丹尼斯·泊松（1781-1840）提出，与他的概率理论一起发表在他的著作《苏拉河畔的调查》中，'''sur la probabilité des jugements en matière criminelle et en matière civile(1837)'''
这项工作通过关注某些随机变量 n (其中包括在给定时间间隔内发生的离散事件(有时称为“事件”或“到达事件”)的数量)来推断某一国家的错误定罪数量。这个结果早在1711年就已经在《亚伯拉罕·棣莫弗》给出了。在 Ludis 举行的猜测活动。这使它成为斯蒂格勒定律的一个例子，也使一些作者提出，''' 泊松分佈Poisson distribution '''应该以德莫伊弗雷的名字命名。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

In 1860, [[Simon Newcomb]] fitted the Poisson distribution to the number of stars found in a unit of space.{{r|Newcomb1860}}

In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.

1860年，Simon Newcomb 将''' 泊松分佈Poisson distribution '''与天文台一个空间单位中发现的恒星数量进行了比较。

A further practical application of this distribution was made by [[Ladislaus Bortkiewicz]] in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks;{{r|vonBortkiewitsch1898|p=23-25}} this experiment introduced the Poisson distribution to the field of [[reliability engineering]].

A further practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to the field of reliability engineering.

这种分布的进一步实际应用是在1898年，当时拉迪斯劳斯·博特基威茨被赋予任务调查普鲁士军队中被马踢意外杀死的士兵人数; 这个实验将''' 泊松分佈Poisson distribution '''引入可靠性技术领域。

--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”

== See also又及 ==

{{div col |colwidth = 18em }}

* [[Compound Poisson distribution]]

* [[Conway–Maxwell–Poisson distribution]]

* [[Erlang distribution]]

* [[Hermite distribution]]

* [[Index of dispersion]]

* [[Negative binomial distribution]]

* [[Poisson clumping]]

* [[Poisson point process]]

* [[Poisson regression]]

* [[Poisson sampling]]

* [[Poisson wavelet]]

* [[Queueing theory]]

* [[Renewal theory]]

* [[Robbins lemma]]

* [[Skellam distribution]]

* [[Tweedie distribution]]

* [[Zero-inflated model]]

* [[Zero-truncated Poisson distribution]]

{{div col end}}

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|title=Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés

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|trans-title=Research on the Probability of Judgments in Criminal and Civil Matters

| trans-title = 刑事和民事案件判决的可能性研究

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{{refend}}

{{-}}

{{ProbDistributions|discrete-infinite}}

{{Authority control}}

[[Category:Poisson distribution| ]]

[[Category:Articles with example pseudocode]]

Category:Articles with example pseudocode

类别: 带有伪代码示例的文章

[[Category:Conjugate prior distributions]]

Category:Conjugate prior distributions

范畴: 共轭先验分布

[[Category:Factorial and binomial topics]]

Category:Factorial and binomial topics

类别: 阶乘和二项式主题

[[Category:Infinitely divisible probability distributions]]

Category:Infinitely divisible probability distributions

类别: 无限可分的概率分布

<noinclude>

This page was moved from [[wikipedia:en:Poisson distribution]]. Its edit history can be viewed at [[泊松分布/edithistory]]</noinclude>

[[Category:待整理页面]]

概率分布

2021-01-11T11:30:51Z

Fairywang：

本词条由Ryan初步翻译

{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}

{{other uses|Distribution (disambiguation){{!}}Distribution}}

{{Off topic|date=August 2020|Random variable|talksectionheading}}

In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].<ref name=":02">{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}</ref><ref>{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}</ref> It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).<ref name=":1">{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}</ref>

In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

在'''概率论 Probability Theory'''和'''统计学 Statistics'''中，''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。”一句改为“''' 概率分布 Probability Distribution'''是一个数学函数，它给出一个试验不同可能结果出现的概率。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。”一句中“(样本空间的子集)”改为“（事件即样本空间的子集）”。

For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X'' {{=}} heads}}, and 0.5 for {{math|''X'' {{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.<ref name="ross">{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}</ref>

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.

例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。”一句改为“例如，如果使用随机变量{{mvar|X}}来表示掷硬币（“试验”）的结果，那么{{mvar|X}}的概率分布是：{{math|''X'' {{=}} 正面}}的概率值为0.5，{{math|''X'' {{=}} 反面}}的概率值为0.5（假设硬币是公平的）。”

A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values. For example, the sample space of a coin flip would be {{math|{heads, tails} }}.

A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values. For example, the sample space of a coin flip would be .

概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。”一句改为“它可能是实数的集合或向量的集合，也可能是非数值的集合。例如，抛硬币的样本空间是{正面，反面}。”

Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].<ref name="ross" /><ref>{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}</ref> The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.

Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

概率分布一般分为两类。''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。''' 正态分布 Normal Distribution'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况”一句中“一组可能的结果”改为“可能结果的集合”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况”一句中“一组可以在一个连续的范围内取值的结果”改为“可能结果集在连续范围内取值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。”一句改为“更复杂的试验，例如那些涉及用连续时间定义的随机过程的试验，可能需要使用更具有一般性的概率测度。”

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为''' 单变量 Univariate'''，而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''”一句中“二维或更多向量空间”改为“二维或更多维向量空间”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“单变量分布给出了单个随机变量取不同替代值的概率”一句中“不同替代值”改为“不同可能值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。”一句改为“多变量分布（联合概率分布）给出了一个随机向量（两个或多个随机变量组成的列表）各种取值组合的概率。”

==Introduction 简介==

[[File:Dice Distribution (bar).svg|thumb|250px|right|
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]]. For example, the figure shows that ''p''(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as ''P''(''S'' > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]

概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s > 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布”一句中“指定”改为“列出了”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“Pmf 允许计算事件的概率”一句中“允许”改为“可用于”。

To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] <math>p</math> assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个'''概率质量函数 Probability Mass Function''' P就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是

<math>p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“6个值中的每一个的概率为1/6”一句中“6个值”改为“1-6这六个值”。

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率”一句改为“通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件（例如区间）才有正的概率”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98%”一句改为“人们可能会要求介于490克至510克之间的“500克”包装出现的概率不低于98%”。

Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

连续概率分布可以用几种方法来描述。'''概率密度函数 Probability Density Function'''描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。”一句改为“并且结果落在给定区间内的概率可以通过对该区间的概率密度函数进行积分来计算。”

[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“它是概率密度曲线下面的区域”一句中“下面的区域”改为“下方的面积”。

The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity <math>\infty</math> to <math>x</math> as described by the picture to the right.<ref>{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}</ref>

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.

累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数指标描述了随机变量不大于给定值的概率”一句中，删除“指标”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算”一句改为“结果落在给定区间内的概率可以通过累积分布函数在区间端点的值之间的差来计算。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数是从负无穷到 x 的概率密度函数下面的区域”一句中“下面的区域”改为“下方的面积”。

[[File:Standard deviation diagram.svg|right|thumb|250px|
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“也称为高斯或钟形曲线”一句中“高斯”改为“高斯分布”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“值间隔的概率对应于曲线下面积”一句中“值间隔的概率”改为“区间里的值所表示的概率”。

== Terminology 术语==

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.<ref name=":02" />

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X > a, X Functions for discrete variables 离散变量函数'''===

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。

*'''Probability function''': describes the probability distribution of a discrete random variable.
''' 概率函数 Probability Function'''：描述离散随机变量的概率分布。

*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.
''' 概率质量函数（pmf Probability Mass Function'''：给出离散随机变量等于某个值的概率的函数。

*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.
''' 频率分布 Frequency Distribution'''：显示样本中各种结果的频率的表格。

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.
''' 相对频率分布 Relative Frequency Distribution'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“多个结果”改为“结果数”。

*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.
'''离散概率分布函数 Discrete Probability Distribution Function'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“整个人群”改为“整个总体”。

*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a discrete random variable.
''' 累积分布函数 Cumulative distribution function'''：该函数评估离散随机变量X取小于或等于x的值的概率。

*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.
''' 分类分布 Categorical Distribution'''：适用于具有有限值集的离散随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“适用于具有有限值集”改为“取值结果为有限集合”。

=== ''' Functions for continuous variables 连续变量函数'''===

* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.
''' 概率密度函数（pdf）Probability Density function'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“在样本空间（随机变量可能取值的集合）中任意给定样本（或样本点），此样本在该函数上的取值可以被解释成给出了随机变量取值等于此样本的相对可能性 relative likelihood”。

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

* '''Continuous probability distribution function''': most often reserved for continuous random variables.
'''连续概率分布函数 Continuous Probability Distribution Function'''：最常保留的连续随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“保留的”改为“用于”。

* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for continuous variable.
''' 累积分布函数 Cumulative distribution function'''：评估连续变量X取小于或等于x的值的概率的函数。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“评估”改为“计算”；“取小于或等于x的值”改为“取值小于或等于x”。

===Basic terms 基本术语===

* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“模式”改为“众数”。

* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“支持”改为“支撑集”；“一组值”改为“值的集合”；“有时表示为R_ {X}”改为“它的支撑集有时表示为R_ {X}”。

* '''Tail''':<ref name='tail'>More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]</ref> the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form <math>X > a</math>, <math>X < b</math> or a union thereof.
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X> a，X <b或它们的并集。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句前一半改为“尾部：当pmf或pdf相对较低时，靠近随机变量边界的区域。”

*'''Head''':<ref name='tail' /> the region where the pmf or pdf is relatively high. Usually has the form <math>a < X < b</math>.
头部：pmf或pdf较高的区域。通常具有a <X <b的形式

* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“或其连续类似物”改为“或连续随机变量的类似取值。”

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此段应出现在下文，此处重复。

*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“中位数：此值使得小于中位数的取值集合和大于中位数的取值集合各自的概率都不大于二分之一。”

*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“方差：关于均值的pmf或pdf的二阶矩；度量分布离散性的重要指标。”

* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.
标准偏差：方差的平方根，因此是色散的另一种度量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“标准差：方差的平方根，因此是度量离散程度的另一指标。”

* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。

*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third [[standardized moment]] of the distribution.
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“分布的第三个标准化时刻”改为“分布的三阶矩”。

*'''[[Kurtosis]]''': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“峰度：pmf或pdf尾部“胖瘦”的量度。分布的四阶矩。”。

==''' Discrete probability distribution 离散概率分布'''==

[[File:Discrete probability distrib.svg|right|thumb|
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]

[[File:Discrete probability distribution.svg|right|thumb|
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]

[[File:Normal probability distribution.svg|right|thumb|
图6:... of a continuous probability distribution, ...连续概率分布]]

[[File:Mixed probability distribution.svg|right|thumb|
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图4图注未译，应为“离散概率分布的概率质量函数。单元集{1}, {3}, 和{7}的概率分别为0.2, 0.5, 0.3。不包含这些点的集合的概率是0。”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图5图注改为“离散概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图6图注改为“连续概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图7图注改为“既有连续部分又有离散部分的分布的累积分布函数”。

A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.<ref>{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}</ref> For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if <math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1”一句中“无限大”改为“可数无穷大”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】含公式句中“如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2”改为“如果对于''n'' = 1, 2, ...有<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math>”。

Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].<ref name=":1" /> Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“统计建模中使用的众所周知的离散概率分布包括”中“使用的众所周知的”改为“常用的”。

When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。

===Measure theoretic formulation 测量理论公式===

A [[measurable function]] <math>X \colon A \to B </math> between a [[probability space]] <math>(A, \mathcal A, P)</math> and a [[measurable space]] <math>(B, \mathcal B) </math> is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of <math>X</math> means that the pre-images of singleton sets are measurable, i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> for all <math>b \in B</math>.
The latter requirement induces a [[probability mass function]] <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. Since the pre-images of disjoint sets
are disjoint,
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
This recovers the definition given above.

一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。”一句改为“在概率空间<math>(A, \mathcal A, P)</math> 和可测空间之间的一个可测函数 <math>X \colon A \to B </math>称为离散随机变量，它的像是一个可数集合。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.”一句改为“在这种情况下<math>X</math>的可测量性意味着单元集的原像是可测量的，即对于所有的<math>b \in B</math>，有 <math>X^{-1}(\{b\}) \in \mathcal A</math> 。”

后者需要包括概率质量函数 <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. 由于不相交集的原像不相交
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
这包含了上面所提到的定义

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“后者需要”改为“后一个必要条件”。

===''' 累积分布函数 Cumulative distribution function'''===

Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.

与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“密集的实数集”改为“实数的稠密集”。

===Delta-function representation 三角函数表示===

Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.<ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}</ref>

因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“三角函数表示”改为“&delta函数表示”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数”改为“离散概率分布通常表示为包含狄拉克&delta函数的广义概率密度函数”。

===Indicator-function representation 指标功能表示===

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“指标功能表示”改为“指示函数表示”。
For a discrete random variable ''X'', let ''u''0, ''u''1, ... be the values it can take with non-zero probability. Denote
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示
:<math>\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“表示”改为“令”。
这些是不相交的集合，对于这样的集合：

:<math>P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>

因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“写入为”改为“写为”。

:<math>X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>

except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。

==Continuous probability distribution 连续概率分布==

{{See also|Probability density function}}

A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.<ref>{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}</ref> They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。”一句改为“连续概率分布是支撑集为不可数集的概率分布，如实数轴上的区间。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]”一句中，“支撑的每个子集”改为“支撑集每个子集”。

A random variable <math>X</math> has a continuous probability distribution if there is a function <math>f: \mathbb{R} \rightarrow [0, \infty)</math> such that for each interval <math>I \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> For example, if <math>I = [a, b]</math> then we would have:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

一个随机变量<math>X</math> 有一个连续的概率分布，如果这有一个函数<math>f: \mathbb{R} \rightarrow [0, \infty)</math> 对于每一个区间<math>I \subset \mathbb{R}</math> <math>I</math>的概率 <math>X</math>是<math>f</math> 在 <math>I</math>上的积分.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> 例如, 如果<math>I = [a, b]</math> 可得到:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如”前一句中，将“如果”以后内容作为前半句，“对于”前加“使得”二字，“<math>I \subset \mathbb{R}</math>”后加逗号，“<math>I</math>的概率 <math>X</math>”改为“随机变量 <math>X</math>属于<math>I</math>的概率”，“一个随机变量<math>X</math> 有一个连续的概率分布”改为“则称这个随机变量<math>X</math> 有一个连续的概率分布”并置于后半句。

In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as

:<math>F(x) = \operatorname{P}\left[-\infty < X \le x\right] = \int_{-\infty}^x f(x) \, dx</math>

特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“特别是”改为“特别地”，“其累积密度函数定义为”下面应有上方那行公式。

which, by this definition, has the properties:
根据定义有以下一些性质

<ul>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>

<li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>\lim_{x \rightarrow -\infty} F(x) = 0</math> and <math>\lim_{x \rightarrow \infty} F(x) = 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>P(a \le X < b) = F(b) - F(a)</math>; and</li>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is continuous (due to the [[Riemann integral]] properties).</li>

</ul>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】以上性质没有翻译。第一行应为“<math>F(x)</math>是非减的”；第三行"and"改为“和”；第四行"and"改为“以及”；第五行应为“由于黎曼积分的性质，<math>F(x)</math>连续”

It is also possible to think in the opposite direction, which allows more flexibility. Say <math>F(x)</math> is a function that satisfies all but the last of the properties above, then <math>F</math> represents the cumulative density function for some random variable: a discrete random variable if <math>F</math> is a step function, and a continuous random variable otherwise.<ref>See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.</ref> This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“假设 f(x)是满足上述所有性质的函数”改为“假设 f(x)是满足上述除最后一条外其他所有性质的函数”，“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If <math>X</math> is such an absolutely continuous random variable, then it has a [[probability density function]] <math>f(x)</math>, and its probability of falling into a Lebesgue-measurable set <math>A \subset \mathbb{R}</math> is:

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

where <math>\mu</math> is the Lebesgue measure.

对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

这里，mu是Lebesgue度量

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“对于实线的更多任意子集，通常有必要对上述定义进行概括”中“实线”改为“实数轴”。“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones <math>\mu</math> such that <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].

关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“既不是绝对连续也不是离散的，也不是它们的混合。没有密度。”改为“这种奇异分布既不是绝对连续，也不是离散的，也不是它们的混合，同时也没有密度。”

== ''' [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义'''==

{{Main|Probability space|Probability measure}}

In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}</ref>

在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足<math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>的概率量度

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在概率论的度量理论形式化中”一句中“度量理论”改为“测度论”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）”一句改为“随机变量定义为从概率空间<math>(\Omega, \mathcal{F}, \mathbb{P})</math>到测度空间<math>(\mathcal{X},\mathcal{A})</math>的测度函数<math>(\mathcal{X},\mathcal{A})</math>。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P”一句改为“假设这种形式的事件概率<math>\{\omega\in\Omega\mid X(\omega)\in A\}</math>满足满足柯尔莫哥洛夫的概率公理，则X的概率分布为X的pushforward measure<math>X_*\mathbb{P}</math>”。

==Random number generation 随机数生成==
{{Main|Pseudo-random number sampling}}

Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}</ref>

大多数算法基于''' 伪随机数生成器 Pseudorandom Number Generator'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“半开间隔”改为“半开区间”。

For example, suppose <math>U</math> has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some <math>0 , we define

<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

so that

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

因此

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“因此”改为“使得”。

This random variable X has a Bernoulli distribution with parameter <math>p</math>.<ref name=":0"/> Note that this is a transformation of discrete random variable.
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“该随机变量X具有参数的伯努利分布p”改为“该随机变量X具有以p为参数的伯努利分布”。

For a distribution function <math>F</math> of a continuous random variable, a continuous random variable must be constructed. <math>F^{inv}</math>, an inverse function of <math>F</math>, relates to the uniform variable <math>U</math>:

<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

F或连续随机变量的分布函数F，必须构造连续随机变量。 <math>F^{inv}</math>，F的反函数，涉及均匀变量U：
<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F或连续随机变量的分布函数F”中“F或”改为“对于”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F的反函数，涉及均匀变量U”改为“F反函数<math>F^{inv}</math>与均匀变量U有以下关系：”。

For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.
例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。

<math>\begin{align}
F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\ &\Leftrightarrow e^{-\lambda x } = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)
\end{align}</math>

所以<math>F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)</math> 并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后随机变量 <math>X</math> 被定义为 <math>X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)</math>. 这里有一个指数分布 <math>\lambda</math>.<ref name=":0" />
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】"并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后"改为“如果 <math>U</math> 服从<math>(0,1)</math> 分布, 且”
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。

== Common probability distributions and their applications 共同概率分布及其应用==
{{Main list|List of probability distributions}}
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“共同概率分布及其应用”改为“常见概率分布及其应用”
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

概率分布和内蕴其中的随机变量的概念是概率论和统计学领域的基础。几乎任何可以在群体中测量的值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量都存在一定的固有误差；在物理学中，从气体动力学理论到基本粒子的量子力学描述都用概率论来描述。出于各种原因，简单数字通常很难表述一个量量，而用概率分布表述通常更合适。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“简单数字通常很难表述一个量量”改为“简单数字通常不足以表述一个量”
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)
以下列出了一些最常见的概率分布，按与之相关的过程类型进行分组。更完整的内容请参见概率分布列表，该列表按研究对象的性质（离散，连续，多元等）进行分组。

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].
下面所有的单变量分布都是单峰的。也就是说，假设值聚集在单个值周围。实际上，实际观测量可能会聚集在多个值附近，可以使用混合分布对这种量进行建模。

=== '''Linear growth (e.g. errors, offsets) 线性增长 '''===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“线性增长”改为“线性增长（例如，错误，偏差等）”
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution
''' 正态分布（高斯分布 Normal Distribution'''，适用于单个变量；是最常用的连续分布

=== ''' Exponential growth (e.g. prices, incomes, populations) 指数增长'''===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“指数增长”改为“指数增长（例如，价格，收入，人口）”
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed
'''对数正态分布 Log-normal Distribution'''，适用于对数为正态分布的单个变量

* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution
'''帕累托分布 Pareto Distribution'''，适用于指数为正态分布的单个变量；是幂律分布的原型

=== ''' Uniformly distributed quantities 数量均匀分布'''===

* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)
''' 离散均匀分布 Discrete Uniform Distributed'''，用于有限的一组值（例如，均匀骰子的结果）

* [[Continuous uniform distribution]], for continuously distributed values
'''连续均匀分布 Continuous Uniform Distributed'''，用于连续分布的值

=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（给定的概率的是或否事件===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“给定的概率的是或否事件”改为“给定了概率的是或否事件”
* Basic distributions 基本分布:
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
''' 伯努利分布 Bernoulli Distribution'''，用于描述单个伯努利试验的结果（例如成功/失败，是/否）

** [[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences
'''二项式分布 Binomial Distribution '''，用于描述给定总数的前提下相互独立变量的成功次数（例如，成功，赞成票等）

** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
'''负二项分布 Negative Binomial Distribution'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数

** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]
'''几何分布 Geometric Distribution'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；是负二项分布的特殊情况

* Related to sampling schemes over a finite population 与有限人口抽样方案有关:

** [[Hypergeometric distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]
''' 超几何分布 Hypergeometric Distribution'''，描述“成功”的次数（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换

** [[Beta-binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the "opposite" of [[sampling without replacement]])
''' 贝塔二项式分布 Beta-binomial Distribution'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换））

=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===

* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]
针对单个分类结果的'''分类分布 Categorical Distribution'''（例如，调查中的是/否/也许）；伯努利分布的一般化

* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]
给定总结果的固定数量，针对每种类别结果的数量的'''多项式分布 Multinomial Distribution'''；二项式分布的一般化

* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]
'''多元超几何分布 Multivariate Distribution'''，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化

=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===

* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
''' 泊松分布 Poisson Distribution'''，用于给定时间段内泊松型事件的发生次数

* [[Exponential distribution]], for the time before the next Poisson-type event occurs
'''指数分布 Exponential Distribution'''，在下一次泊松型事件发生之前的时间

* [[Gamma distribution]], for the time before the next k Poisson-type events occur
'''伽马分布 Gamma Distribution'''，在接下来的k个泊松型事件发生之前的时间

=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===

* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
'''瑞利分布 Rayleigh Distribution'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。

* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.
'''莱斯分布 Rice Distribution'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。

=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===

* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])
'''卡方分布 Chi-squared Distribution'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）

* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
'''学生t分布 Student‘s t Distribution'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）

* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])
'''F-分布 F-Distribution'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论

=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===
{{Main|Conjugate prior}}

* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
'''Beta分布 Beta Distribution'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭

* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。

* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]
'''Dirichlet分布 Dirichlet Distribution'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化

*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]<ref>{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}</ref>
'''Wishart分布 Wishart Distribution'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化

=== Some specialized applications of probability distributions 概率分布的一些特殊应用===

* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。”改为“在自然语言处理中，使用的高速缓存语言模型和其他统计语言模型，通过概率分布，为特定单词和单词序列的出现分配概率。”
* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为<math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。

* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。

* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。

==See also 另请参见==
{{Portal|Mathematics}}
*[[Copula (statistics)]]
Copula（统计数据）
* [[Empirical probability]]
经验概率
* [[Histogram]]
直方图
* [[Likelihood function]]
似然函数
* [[Kirkwood approximation]]
柯克伍德近似
* [[Moment-generating function]]
瞬时产生功能
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]
Riemann–Stieltjes积分＃在概率论中的应用
*[[Pairwise independence]]
成对独立

=== Lists 清单===

* [[List of probability distributions]]
概率分布的清单
* [[List of statistical topics]]
统计学话题的清淡
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“统计学话题的清淡”改为“统计学话题的清单”
=== Probability distributions 概率分布 ===

* [[Conditional probability distribution]]
条件概率分布
* [[Joint probability distribution]]
联合概率分布
* [[Quasiprobability distribution]]
拟概率分布

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin|}}
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}
{{refend}}

==External links==
{{commons|Probability distribution|Probability distribution}}
*{{springer|title=Probability distribution|id=p/p074900}}
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions], Gavin E. Crooks.

概率分布

2021-01-11T11:27:29Z

Fairywang：

本词条由Ryan初步翻译

{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}

{{other uses|Distribution (disambiguation){{!}}Distribution}}

{{Off topic|date=August 2020|Random variable|talksectionheading}}

In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].<ref name=":02">{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}</ref><ref>{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}</ref> It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).<ref name=":1">{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}</ref>

In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

在'''概率论 Probability Theory'''和'''统计学 Statistics'''中，''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。”一句改为“''' 概率分布 Probability Distribution'''是一个数学函数，它给出一个试验不同可能结果出现的概率。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。”一句中“(样本空间的子集)”改为“（事件即样本空间的子集）”。

For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X'' {{=}} heads}}, and 0.5 for {{math|''X'' {{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.<ref name="ross">{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}</ref>

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.

例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。”一句改为“例如，如果使用随机变量{{mvar|X}}来表示掷硬币（“试验”）的结果，那么{{mvar|X}}的概率分布是：{{math|''X'' {{=}} 正面}}的概率值为0.5，{{math|''X'' {{=}} 反面}}的概率值为0.5（假设硬币是公平的）。”

A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values. For example, the sample space of a coin flip would be {{math|{heads, tails} }}.

A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values. For example, the sample space of a coin flip would be .

概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。”一句改为“它可能是实数的集合或向量的集合，也可能是非数值的集合。例如，抛硬币的样本空间是{正面，反面}。”

Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].<ref name="ross" /><ref>{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}</ref> The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.

Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

概率分布一般分为两类。''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。''' 正态分布 Normal Distribution'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况”一句中“一组可能的结果”改为“可能结果的集合”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况”一句中“一组可以在一个连续的范围内取值的结果”改为“可能结果集在连续范围内取值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。”一句改为“更复杂的试验，例如那些涉及用连续时间定义的随机过程的试验，可能需要使用更具有一般性的概率测度。”

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为''' 单变量 Univariate'''，而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''”一句中“二维或更多向量空间”改为“二维或更多维向量空间”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“单变量分布给出了单个随机变量取不同替代值的概率”一句中“不同替代值”改为“不同可能值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。”一句改为“多变量分布（联合概率分布）给出了一个随机向量（两个或多个随机变量组成的列表）各种取值组合的概率。”

==Introduction 简介==

[[File:Dice Distribution (bar).svg|thumb|250px|right|
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]]. For example, the figure shows that ''p''(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as ''P''(''S'' > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]

概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s > 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布”一句中“指定”改为“列出了”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“Pmf 允许计算事件的概率”一句中“允许”改为“可用于”。

To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] <math>p</math> assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个'''概率质量函数 Probability Mass Function''' P就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是

<math>p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“6个值中的每一个的概率为1/6”一句中“6个值”改为“1-6这六个值”。

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率”一句改为“通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件（例如区间）才有正的概率”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98%”一句改为“人们可能会要求介于490克至510克之间的“500克”包装出现的概率不低于98%”。

Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

连续概率分布可以用几种方法来描述。'''概率密度函数 Probability Density Function'''描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。”一句改为“并且结果落在给定区间内的概率可以通过对该区间的概率密度函数进行积分来计算。”

[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“它是概率密度曲线下面的区域”一句中“下面的区域”改为“下方的面积”。

The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity <math>\infty</math> to <math>x</math> as described by the picture to the right.<ref>{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}</ref>

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.

累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数指标描述了随机变量不大于给定值的概率”一句中，删除“指标”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算”一句改为“结果落在给定区间内的概率可以通过累积分布函数在区间端点的值之间的差来计算。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数是从负无穷到 x 的概率密度函数下面的区域”一句中“下面的区域”改为“下方的面积”。

[[File:Standard deviation diagram.svg|right|thumb|250px|
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“也称为高斯或钟形曲线”一句中“高斯”改为“高斯分布”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“值间隔的概率对应于曲线下面积”一句中“值间隔的概率”改为“区间里的值所表示的概率”。

== Terminology 术语==

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.<ref name=":02" />

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X > a, X Functions for discrete variables 离散变量函数'''===

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。

*'''Probability function''': describes the probability distribution of a discrete random variable.
''' 概率函数 Probability Function'''：描述离散随机变量的概率分布。

*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.
''' 概率质量函数（pmf Probability Mass Function'''：给出离散随机变量等于某个值的概率的函数。

*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.
''' 频率分布 Frequency Distribution'''：显示样本中各种结果的频率的表格。

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.
''' 相对频率分布 Relative Frequency Distribution'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“多个结果”改为“结果数”。

*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.
'''离散概率分布函数 Discrete Probability Distribution Function'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“整个人群”改为“整个总体”。

*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a discrete random variable.
''' 累积分布函数 Cumulative distribution function'''：该函数评估离散随机变量X取小于或等于x的值的概率。

*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.
''' 分类分布 Categorical Distribution'''：适用于具有有限值集的离散随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“适用于具有有限值集”改为“取值结果为有限集合”。

=== ''' Functions for continuous variables 连续变量函数'''===

* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.
''' 概率密度函数（pdf）Probability Density function'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“在样本空间（随机变量可能取值的集合）中任意给定样本（或样本点），此样本在该函数上的取值可以被解释成给出了随机变量取值等于此样本的相对可能性 relative likelihood”。

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

* '''Continuous probability distribution function''': most often reserved for continuous random variables.
'''连续概率分布函数 Continuous Probability Distribution Function'''：最常保留的连续随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“保留的”改为“用于”。

* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for continuous variable.
''' 累积分布函数 Cumulative distribution function'''：评估连续变量X取小于或等于x的值的概率的函数。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“评估”改为“计算”；“取小于或等于x的值”改为“取值小于或等于x”。

===Basic terms 基本术语===

* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“模式”改为“众数”。

* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“支持”改为“支撑集”；“一组值”改为“值的集合”；“有时表示为R_ {X}”改为“它的支撑集有时表示为R_ {X}”。

* '''Tail''':<ref name='tail'>More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]</ref> the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form <math>X > a</math>, <math>X < b</math> or a union thereof.
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X> a，X <b或它们的并集。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句前一半改为“尾部：当pmf或pdf相对较低时，靠近随机变量边界的区域。”

*'''Head''':<ref name='tail' /> the region where the pmf or pdf is relatively high. Usually has the form <math>a < X < b</math>.
头部：pmf或pdf较高的区域。通常具有a <X <b的形式

* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“或其连续类似物”改为“或连续随机变量的类似取值。”

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此段应出现在下文，此处重复。

*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“中位数：此值使得小于中位数的取值集合和大于中位数的取值集合各自的概率都不大于二分之一。”

*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“方差：关于均值的pmf或pdf的二阶矩；度量分布离散性的重要指标。”

* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.
标准偏差：方差的平方根，因此是色散的另一种度量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“标准差：方差的平方根，因此是度量离散程度的另一指标。”

* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。

*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third [[standardized moment]] of the distribution.
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“分布的第三个标准化时刻”改为“分布的三阶矩”。

*'''[[Kurtosis]]''': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“峰度：pmf或pdf尾部“胖瘦”的量度。分布的四阶矩。”。

==''' Discrete probability distribution 离散概率分布'''==

[[File:Discrete probability distrib.svg|right|thumb|
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]

[[File:Discrete probability distribution.svg|right|thumb|
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]

[[File:Normal probability distribution.svg|right|thumb|
图6:... of a continuous probability distribution, ...连续概率分布]]

[[File:Mixed probability distribution.svg|right|thumb|
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图4图注未译，应为“离散概率分布的概率质量函数。单元集{1}, {3}, 和{7}的概率分别为0.2, 0.5, 0.3。不包含这些点的集合的概率是0。”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图5图注改为“离散概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图6图注改为“连续概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图7图注改为“既有连续部分又有离散部分的分布的累积分布函数”。

A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.<ref>{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}</ref> For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if <math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1”一句中“无限大”改为“可数无穷大”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】含公式句中“如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2”改为“如果对于''n'' = 1, 2, ...有<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math>”。

Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].<ref name=":1" /> Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“统计建模中使用的众所周知的离散概率分布包括”中“使用的众所周知的”改为“常用的”。

When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。

===Measure theoretic formulation 测量理论公式===

A [[measurable function]] <math>X \colon A \to B </math> between a [[probability space]] <math>(A, \mathcal A, P)</math> and a [[measurable space]] <math>(B, \mathcal B) </math> is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of <math>X</math> means that the pre-images of singleton sets are measurable, i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> for all <math>b \in B</math>.
The latter requirement induces a [[probability mass function]] <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. Since the pre-images of disjoint sets
are disjoint,
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
This recovers the definition given above.

一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。”一句改为“在概率空间<math>(A, \mathcal A, P)</math> 和可测空间之间的一个可测函数 <math>X \colon A \to B </math>称为离散随机变量，它的像是一个可数集合。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.”一句改为“在这种情况下<math>X</math>的可测量性意味着单元集的原像是可测量的，即对于所有的<math>b \in B</math>，有 <math>X^{-1}(\{b\}) \in \mathcal A</math> 。”

后者需要包括概率质量函数 <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. 由于不相交集的原像不相交
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
这包含了上面所提到的定义

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“后者需要”改为“后一个必要条件”。

===''' 累积分布函数 Cumulative distribution function'''===

Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.

与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“密集的实数集”改为“实数的稠密集”。

===Delta-function representation 三角函数表示===

Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.<ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}</ref>

因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“三角函数表示”改为“&delta函数表示”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数”改为“离散概率分布通常表示为包含狄拉克&delta函数的广义概率密度函数”。

===Indicator-function representation 指标功能表示===

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“指标功能表示”改为“指示函数表示”。
For a discrete random variable ''X'', let ''u''0, ''u''1, ... be the values it can take with non-zero probability. Denote
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示
:<math>\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“表示”改为“令”。
这些是不相交的集合，对于这样的集合：

:<math>P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>

因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“写入为”改为“写为”。

:<math>X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>

except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。

==Continuous probability distribution 连续概率分布==

{{See also|Probability density function}}

A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.<ref>{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}</ref> They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。”一句改为“连续概率分布是支撑集为不可数集的概率分布，如实数轴上的区间。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]”一句中，“支撑的每个子集”改为“支撑集每个子集”。

A random variable <math>X</math> has a continuous probability distribution if there is a function <math>f: \mathbb{R} \rightarrow [0, \infty)</math> such that for each interval <math>I \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> For example, if <math>I = [a, b]</math> then we would have:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

一个随机变量<math>X</math> 有一个连续的概率分布，如果这有一个函数<math>f: \mathbb{R} \rightarrow [0, \infty)</math> 对于每一个区间<math>I \subset \mathbb{R}</math> <math>I</math>的概率 <math>X</math>是<math>f</math> 在 <math>I</math>上的积分.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> 例如, 如果<math>I = [a, b]</math> 可得到:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如”前一句中，将“如果”以后内容作为前半句，“对于”前加“使得”二字，“<math>I \subset \mathbb{R}</math>”后加逗号，“<math>I</math>的概率 <math>X</math>”改为“随机变量 <math>X</math>属于<math>I</math>的概率”，“一个随机变量<math>X</math> 有一个连续的概率分布”改为“则称这个随机变量<math>X</math> 有一个连续的概率分布”并置于后半句。

In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as

:<math>F(x) = \operatorname{P}\left[-\infty < X \le x\right] = \int_{-\infty}^x f(x) \, dx</math>

特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“特别是”改为“特别地”，“其累积密度函数定义为”下面应有上方那行公式。

which, by this definition, has the properties:
根据定义有以下一些性质

<ul>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>

<li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>\lim_{x \rightarrow -\infty} F(x) = 0</math> and <math>\lim_{x \rightarrow \infty} F(x) = 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>P(a \le X < b) = F(b) - F(a)</math>; and</li>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is continuous (due to the [[Riemann integral]] properties).</li>

</ul>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】以上性质没有翻译。第一行应为“<math>F(x)</math>是非减的”；第三行"and"改为“和”；第四行"and"改为“以及”；第五行应为“由于黎曼积分的性质，<math>F(x)</math>连续”

It is also possible to think in the opposite direction, which allows more flexibility. Say <math>F(x)</math> is a function that satisfies all but the last of the properties above, then <math>F</math> represents the cumulative density function for some random variable: a discrete random variable if <math>F</math> is a step function, and a continuous random variable otherwise.<ref>See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.</ref> This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“假设 f(x)是满足上述所有性质的函数”改为“假设 f(x)是满足上述除最后一条外其他所有性质的函数”，“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If <math>X</math> is such an absolutely continuous random variable, then it has a [[probability density function]] <math>f(x)</math>, and its probability of falling into a Lebesgue-measurable set <math>A \subset \mathbb{R}</math> is:

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

where <math>\mu</math> is the Lebesgue measure.

对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

这里，mu是Lebesgue度量

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“对于实线的更多任意子集，通常有必要对上述定义进行概括”中“实线”改为“实数轴”。“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones <math>\mu</math> such that <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].

关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“既不是绝对连续也不是离散的，也不是它们的混合。没有密度。”改为“这种奇异分布既不是绝对连续，也不是离散的，也不是它们的混合，同时也没有密度。”

== ''' [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义'''==

{{Main|Probability space|Probability measure}}

In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}</ref>

在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足<math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>的概率量度

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在概率论的度量理论形式化中”一句中“度量理论”改为“测度论”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）”一句改为“随机变量定义为从概率空间<math>(\Omega, \mathcal{F}, \mathbb{P})</math>到测度空间<math>(\mathcal{X},\mathcal{A})</math>的测度函数<math>(\mathcal{X},\mathcal{A})</math>。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P”一句改为“假设这种形式的事件概率<math>\{\omega\in\Omega\mid X(\omega)\in A\}</math>满足满足柯尔莫哥洛夫的概率公理，则X的概率分布为X的pushforward measure<math>X_*\mathbb{P}</math>”。

==Random number generation 随机数生成==
{{Main|Pseudo-random number sampling}}

Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}</ref>

大多数算法基于''' 伪随机数生成器 Pseudorandom Number Generator'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“半开间隔”改为“半开区间”。

For example, suppose <math>U</math> has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some <math>0 , we define

<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

so that

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

因此

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“因此”改为“使得”。

This random variable X has a Bernoulli distribution with parameter <math>p</math>.<ref name=":0"/> Note that this is a transformation of discrete random variable.
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“该随机变量X具有参数的伯努利分布p”改为“该随机变量X具有以p为参数的伯努利分布”。

For a distribution function <math>F</math> of a continuous random variable, a continuous random variable must be constructed. <math>F^{inv}</math>, an inverse function of <math>F</math>, relates to the uniform variable <math>U</math>:

<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

F或连续随机变量的分布函数F，必须构造连续随机变量。 <math>F^{inv}</math>，F的反函数，涉及均匀变量U：
<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F或连续随机变量的分布函数F”中“F或”改为“对于”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F的反函数，涉及均匀变量U”改为“F反函数<math>F^{inv}</math>与均匀变量U有以下关系：”。

For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.
例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。

<math>\begin{align}
F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\ &\Leftrightarrow e^{-\lambda x } = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)
\end{align}</math>

所以<math>F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)</math> 并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后随机变量 <math>X</math> 被定义为 <math>X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)</math>. 这里有一个指数分布 <math>\lambda</math>.<ref name=":0" />
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】"并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后"改为“如果 <math>U</math> 服从<math>(0,1)</math> 分布, 且”
A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。

== Common probability distributions and their applications 共同概率分布及其应用==
{{Main list|List of probability distributions}}
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“共同概率分布及其应用”改为“常见概率分布及其应用”
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

概率分布和内蕴其中的随机变量的概念是概率论和统计学领域的基础。几乎任何可以在群体中测量的值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量都存在一定的固有误差；在物理学中，从气体动力学理论到基本粒子的量子力学描述都用概率论来描述。出于各种原因，简单数字通常很难表述一个量量，而用概率分布表述通常更合适。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“简单数字通常很难表述一个量量”改为“简单数字通常不足以表述一个量”
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)
以下列出了一些最常见的概率分布，按与之相关的过程类型进行分组。更完整的内容请参见概率分布列表，该列表按研究对象的性质（离散，连续，多元等）进行分组。

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].
下面所有的单变量分布都是单峰的。也就是说，假设值聚集在单个值周围。实际上，实际观测量可能会聚集在多个值附近，可以使用混合分布对这种量进行建模。

=== '''Linear growth (e.g. errors, offsets) 线性增长 '''===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“线性增长”改为“线性增长（例如，错误，偏差等）”
* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution
''' 正态分布（高斯分布 Normal Distribution'''，适用于单个变量；是最常用的连续分布

=== ''' Exponential growth (e.g. prices, incomes, populations) 指数增长'''===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“指数增长”改为“指数增长（例如，价格，收入，人口）”
* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed
'''对数正态分布 Log-normal Distribution'''，适用于对数为正态分布的单个变量

* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution
'''帕累托分布 Pareto Distribution'''，适用于指数为正态分布的单个变量；是幂律分布的原型

=== ''' Uniformly distributed quantities 数量均匀分布'''===

* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)
''' 离散均匀分布 Discrete Uniform Distributed'''，用于有限的一组值（例如，均匀骰子的结果）

* [[Continuous uniform distribution]], for continuously distributed values
'''连续均匀分布 Continuous Uniform Distributed'''，用于连续分布的值

=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（给定的概率的是或否事件===
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“给定的概率的是或否事件”改为“给定了概率的是或否事件”
* Basic distributions 基本分布:
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
''' 伯努利分布 Bernoulli Distribution'''，用于描述单个伯努利试验的结果（例如成功/失败，是/否）

** [[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences
'''二项式分布 Binomial Distribution '''，用于描述给定总数的前提下相互独立变量的成功次数（例如，成功，赞成票等）

** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
'''负二项分布 Negative Binomial Distribution'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数

** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]
'''几何分布 Geometric Distribution'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；是负二项分布的特殊情况

* Related to sampling schemes over a finite population 与有限人口抽样方案有关:

** [[Hypergeometric distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]
''' 超几何分布 Hypergeometric Distribution'''，描述“成功”的次数（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换

** [[Beta-binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the "opposite" of [[sampling without replacement]])
''' 贝塔二项式分布 Beta-binomial Distribution'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换））

=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===

* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]
针对单个分类结果的'''分类分布 Categorical Distribution'''（例如，调查中的是/否/也许）；伯努利分布的一般化

* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]
给定总结果的固定数量，针对每种类别结果的数量的'''多项式分布 Multinomial Distribution'''；二项式分布的一般化

* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]
'''多元超几何分布 Multivariate Distribution'''，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化

=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===

* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
''' 泊松分布 Poisson Distribution'''，用于给定时间段内泊松型事件的发生次数

* [[Exponential distribution]], for the time before the next Poisson-type event occurs
'''指数分布 Exponential Distribution'''，在下一次泊松型事件发生之前的时间

* [[Gamma distribution]], for the time before the next k Poisson-type events occur
'''伽马分布 Gamma Distribution'''，在接下来的k个泊松型事件发生之前的时间

=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===

* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
'''瑞利分布 Rayleigh Distribution'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。

* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.
'''莱斯分布 Rice Distribution'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。

=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===

* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])
'''卡方分布 Chi-squared Distribution'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）

* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
'''学生t分布 Student‘s t Distribution'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）

* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])
'''F-分布 F-Distribution'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论

=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===
{{Main|Conjugate prior}}

* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
'''Beta分布 Beta Distribution'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭

* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。

* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]
'''Dirichlet分布 Dirichlet Distribution'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化

*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]<ref>{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}</ref>
'''Wishart分布 Wishart Distribution'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化

=== Some specialized applications of probability distributions 概率分布的一些特殊应用===

* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。

* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为<math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。

* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。

* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。

==See also 另请参见==
{{Portal|Mathematics}}
*[[Copula (statistics)]]
Copula（统计数据）
* [[Empirical probability]]
经验概率
* [[Histogram]]
直方图
* [[Likelihood function]]
似然函数
* [[Kirkwood approximation]]
柯克伍德近似
* [[Moment-generating function]]
瞬时产生功能
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]
Riemann–Stieltjes积分＃在概率论中的应用
*[[Pairwise independence]]
成对独立

=== Lists 清单===

* [[List of probability distributions]]
概率分布的清单
* [[List of statistical topics]]
统计学话题的清淡
=== Probability distributions 概率分布 ===

* [[Conditional probability distribution]]
条件概率分布
* [[Joint probability distribution]]
联合概率分布
* [[Quasiprobability distribution]]
拟概率分布

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin|}}
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}
{{refend}}

==External links==
{{commons|Probability distribution|Probability distribution}}
*{{springer|title=Probability distribution|id=p/p074900}}
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions], Gavin E. Crooks.

概率分布

2021-01-11T11:06:58Z

Fairywang：

本词条由Ryan初步翻译

{{short description|Mathematical function that describes the probability of occurrence of different possible outcomes in an experiment}}

{{other uses|Distribution (disambiguation){{!}}Distribution}}

{{Off topic|date=August 2020|Random variable|talksectionheading}}

In [[probability theory]] and [[statistics]], a '''probability distribution''' is the mathematical [[Function (mathematics)|function]] that gives the probabilities of occurrence of different possible '''outcomes''' for an [[Experiment (probability theory)|experiment]].<ref name=":02">{{Cite book|title=The Cambridge dictionary of statistics|last=Everitt, Brian.|date=2006|publisher=Cambridge University Press|isbn=978-0-511-24688-3|edition=3rd|location=Cambridge, UK|oclc=161828328}}</ref><ref>{{Cite book|title=Basic probability theory|last=Ash, Robert B.|date=2008|publisher=Dover Publications|isbn=978-0-486-46628-6|edition=Dover|location=Mineola, N.Y.|pages=66–69|oclc=190785258}}</ref> It is a mathematical description of a [[Randomness|random]] phenomenon in terms of its [[sample space]] and the [[Probability|probabilities]] of [[Event (probability theory)|events]] (subsets of the sample space).<ref name=":1">{{Cite book|title=Probability and statistics : the science of uncertainty|last=Evans, Michael (Michael John)|date=2010|publisher=W.H. Freeman and Co|others=Rosenthal, Jeffrey S. (Jeffrey Seth)|isbn=978-1-4292-2462-8|edition=2nd|location=New York|pages=38|oclc=473463742}}</ref>

In Probability Theory and Statistics, a Probability Distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

在'''概率论 Probability Theory'''和'''统计学 Statistics'''中，''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 概率分布 Probability Distribution'''是一个给出一个实验不同可能结果出现的概率的数学函数。”一句改为“''' 概率分布 Probability Distribution'''是一个数学函数，它给出一个试验不同可能结果出现的概率。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它是根据''' 样本空间 Sample Space'''和''' 事件概率 Probabilities of Events'''(样本空间的子集)对随机现象的数学描述。”一句中“(样本空间的子集)”改为“（事件即样本空间的子集）”。

For instance, if the [[random variable]] {{mvar|X}} is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of {{mvar|X}} would take the value 0.5 for {{math|''X'' {{=}} heads}}, and 0.5 for {{math|''X'' {{=}} tails}} (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a [[Survey methodology|survey]], etc.<ref name="ross">{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}</ref>

For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X=heads, and 0.5 for X=tails (assuming the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey, etc.

例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。随机现象的例子包括未来某一天的天气状况、一个人的身高、学校中男生的比例、调查结果等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如，如果使用随机变量来表示掷硬币的结果(“实验”) ，那么硬币为正面的概率分布为0.5，反面的值0.5(假设硬币是公平的)。”一句改为“例如，如果使用随机变量{{mvar|X}}来表示掷硬币（“试验”）的结果，那么{{mvar|X}}的概率分布是：{{math|''X'' {{=}} 正面}}的概率值为0.5，{{math|''X'' {{=}} 反面}}的概率值为0.5（假设硬币是公平的）。”

A probability distribution is a mathematical function that has a [[sample space]] as its '''input''', and gives a '''probability''' as its output. The sample space is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] of a random phenomenon being observed; it may be the set of [[real numbers]] or a set of [[vector (mathematics)|vectors]], or it may be a list of non-numerical values. For example, the sample space of a coin flip would be {{math|{heads, tails} }}.

A probability distribution is a mathematical function that has a sample space as its input, and gives a probability as its output. The sample space is the set of all possible outcomes of a random phenomenon being observed; it may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values. For example, the sample space of a coin flip would be .

概率分布是一个数学函数，它的输入是一个样本空间，输出是一个概率。样本空间是所观察到的随机现象的所有可能结果的集合; 它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它可能是一组实数或一组向量，也可能是一组非数值。例如，抛硬币的样本空间是{头，尾}。”一句改为“它可能是实数的集合或向量的集合，也可能是非数值的集合。例如，抛硬币的样本空间是{正面，反面}。”

Probability distributions are generally divided into two classes. A '''discrete probability distribution''' is applicable to the scenarios where the set of possible outcomes is [[discrete probability distribution|discrete]] (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the [[probability mass function]]. On the other hand, '''continuous probability distributions''' are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a [[probability density function]].<ref name="ross" /><ref>{{cite book|author1=DeGroot, Morris H.|author2=Schervish, Mark J.|title=Probability and Statistics|publisher=Addison-Wesley|year=2002}}</ref> The [[normal distribution]] is a commonly encountered continuous probability distribution. More complex experiments, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.

Probability distributions are generally divided into two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss or the roll of a dice), and the probabilities are here encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function. On the other hand, continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In this case, probabilities are typically described by a probability density function. The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

概率分布一般分为两类。''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况，如抛硬币或掷骰子。这里的概率被编码为结果概率的离散列表，称为概率质量函数。另一方面，''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况(例如:实数)，例如某一天的温度。在这种情况下，概率通常由概率密度函数描述。''' 正态分布 Normal Distribution'''是一种常见的连续概率分布。更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 离散概率分布 Discrete Probability Distribution'''适用于一组可能的结果是离散的情况”一句中“一组可能的结果”改为“可能结果的集合”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“''' 连续概率分布 Continuous Probability Distribution'''适用于一组可以在一个连续的范围内取值的结果的情况”一句中“一组可以在一个连续的范围内取值的结果”改为“可能结果集在连续范围内取值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“更复杂的实验，例如那些涉及连续时间定义的随机过程的实验，可能需要使用更一般的概率测度。”一句改为“更复杂的试验，例如那些涉及用连续时间定义的随机过程的试验，可能需要使用更具有一般性的概率测度。”

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called [[Univariate distribution|univariate]], while a distribution whose sample space is a [[vector space]] of dimension 2 or more is called [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a [[joint probability distribution]]) gives the probabilities of a [[random vector]] – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the [[binomial distribution]], the [[hypergeometric distribution]], and the [[normal distribution]]. The [[multivariate normal distribution]] is a commonly encountered multivariate distribution.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

一个一维的样本空间(例如实数、标签列表、有序标签或二进制)的概率分布被称为''' 单变量 Univariate'''，而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''。单变量分布给出了单个随机变量取不同替代值的概率; 联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。重要的和常见的单变量概率分布包括二项分布、超几何分布和正态分布。多变量正态分布是一种常见的联合分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“而样本空间为二维或更多向量空间的分布被称为''' 多变量 Multivariate'''”一句中“二维或更多向量空间”改为“二维或更多维向量空间”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“单变量分布给出了单个随机变量取不同替代值的概率”一句中“不同替代值”改为“不同可能值”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“联合分布给出了一个随机向量的概率——一个由两个或多个随机变量组成的列表——取值的各种组合。”一句改为“多变量分布（联合概率分布）给出了一个随机向量（两个或多个随机变量组成的列表）各种取值组合的概率。”

==Introduction 简介==

[[File:Dice Distribution (bar).svg|thumb|250px|right|
图1:The [[probability mass function]] (pmf) ''p''(''S'') specifies the probability distribution for the sum ''S'' of counts from two [[dice]]. For example, the figure shows that ''p''(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as ''P''(''S'' > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

The [[probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.]]

概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布。例如，图中显示 p (11) = 2/36 = 1/18。Pmf 允许计算事件的概率，如 p (s > 9) = 1/12 + 1/18 + 1/36 = 1/6，以及分布中的所有其他概率。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“概率质量函数(pmf) p(s)指定两个骰子计数总和s的概率分布”一句中“指定”改为“列出了”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“Pmf 允许计算事件的概率”一句中“允许”改为“可用于”。

To define probability distributions for the simplest cases, it is necessary to distinguish between '''discrete''' and '''continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] <math>p</math> assigning a probability to each possible outcome: for example, when throwing a fair [[Dice|die]], each of the six values 1 to 6 has the probability 1/6. The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function p assigning a probability to each possible outcome: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

为了定义最简单的概率分布，有必要区分离散和连续的随机变量。在离散情况下，指定一个'''概率质量函数 Probability Mass Function''' P就足够了，它为每个可能的结果赋予一个概率: 例如，当投掷一个骰子时，6个值中的每一个的概率为1/6。然后将事件的概率定义为满足事件的结果的概率之和; 例如，事件”骰子掷出偶数值”的概率是

<math>p(2) + p(4) + p(6) = 1/6+1/6+1/6=1/2.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“6个值中的每一个的概率为1/6”一句中“6个值”改为“1-6这六个值”。

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs ''exactly'' 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

相比之下，当一个随机变量从一个连续体中取值时，那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率。例如，一个给定的物体重量正好是500克的概率为零，因为随着我们测量仪器精度的提高，正好测量500克的概率趋向于零。然而，在质量控制方面，人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98% ，而这一要求对测量仪器的准确性不太敏感。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“那么通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件，例如间隔，才有正的概率”一句改为“通常情况下，任何单个结果的概率都为零，只有包含无限多个结果的事件（例如区间）才有正的概率”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“人们可能会要求包装在490克至510克之间的“500克”包装的可能性不低于98%”一句改为“人们可能会要求介于490克至510克之间的“500克”包装出现的概率不低于98%”。

Continuous probability distributions can be described in several ways. The [[probability density function]] describes the [[infinitesimal]] probability of any given value, and the probability that the outcome lies in a given interval can be computed by [[Integration (mathematics)|integrating]] the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. The probability that the possible values lie in some fixed interval can be related to the way sums converge to an integral; therefore, continuous probability is based on the definition of an integral.

连续概率分布可以用几种方法来描述。'''概率密度函数 Probability Density Function'''描述了任意给定值的无穷小概率，并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。可能值位于某一固定区间的概率可以与和收敛于积分的方式有关，因此，连续概率是基于积分的定义。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“并且结果在给定区间内的概率可以通过在该区间上积分概率密度函数来计算。”一句改为“并且结果落在给定区间内的概率可以通过对该区间的概率密度函数进行积分来计算。”

[[File:Combined Cumulative Distribution Graphs.png|thumb|455x455px|
图2:On the left is the probability density function. On the right is the cumulative distribution function, which is the area under the probability density curve.
左边是概率密度函数。右边是累积分布函数，它是概率密度曲线下面的区域。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“它是概率密度曲线下面的区域”一句中“下面的区域”改为“下方的面积”。

The [[cumulative distribution function]] describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the [[antiderivative]] of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the [[probability density function]] from minus infinity <math>\infty</math> to <math>x</math> as described by the picture to the right.<ref>{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=Springer|others=Dekking, Michel, 1946-|isbn=978-1-85233-896-1|location=London|oclc=262680588}}</ref>

The cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. The cumulative distribution function is the area under the probability density function from minus infinity \infty to x as described by the picture to the right.

累积分布函数指标描述了随机变量不大于给定值的概率; 结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算。累积分布函数是概率密度函数的反导函数，前提是后者存在。正如右边图片所描述的那样，累积分布函数是从负无穷到 x 的概率密度函数下面的区域。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数指标描述了随机变量不大于给定值的概率”一句中，删除“指标”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“结果在给定区间内的概率可以通过计算区间终点的累积分布函数差来计算”一句改为“结果落在给定区间内的概率可以通过累积分布函数在区间端点的值之间的差来计算。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“累积分布函数是从负无穷到 x 的概率密度函数下面的区域”一句中“下面的区域”改为“下方的面积”。

[[File:Standard deviation diagram.svg|right|thumb|250px|
图3:The [[probability density function]] (pdf) of the [[normal distribution]], also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
正态分布的[概率密度函数(pdf) ，也称为高斯或钟形曲线，是最重要的连续随机分布。如图所示，值间隔的概率对应于曲线下面积。]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“也称为高斯或钟形曲线”一句中“高斯”改为“高斯分布”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图注中“值间隔的概率对应于曲线下面积”一句中“值间隔的概率”改为“区间里的值所表示的概率”。

== Terminology 术语==

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.<ref name=":02" />

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below. the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form X > a, X Functions for discrete variables 离散变量函数'''===

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

用于统计建模的著名离散概率分布包括泊松分佈、伯努利分布、二项分布、几何分布和负二项分布。

*'''Probability function''': describes the probability distribution of a discrete random variable.
''' 概率函数 Probability Function'''：描述离散随机变量的概率分布。

*'''[[Probability mass function|Probability mass function (pmf)]]:''' function that gives the probability that a discrete random variable is equal to some value.
''' 概率质量函数（pmf Probability Mass Function'''：给出离散随机变量等于某个值的概率的函数。

*'''[[Frequency distribution]]''': a table that displays the frequency of various outcomes '''in a sample'''.
''' 频率分布 Frequency Distribution'''：显示样本中各种结果的频率的表格。

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

对于一个离散的随机变量 x，设 u0，u1，... 是它在非零概率情况下可以取的值。表示

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

*'''Relative frequency distribution''': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] i.e. sample size.
''' 相对频率分布 Relative Frequency Distribution'''：一种频率分布，其中每个值均已被样本中的多个结果（即样本大小）除（归一化）。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“多个结果”改为“结果数”。

*'''Discrete probability distribution function''': general term to indicate the way the total probability of 1 is distributed over '''all''' various possible outcomes (i.e. over entire population) for discrete random variable.
'''离散概率分布函数 Discrete Probability Distribution Function'''：通用术语，表示总概率1在离散随机变量的所有各种可能结果（即整个人群）中的分布方式。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“整个人群”改为“整个总体”。

*'''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a discrete random variable.
''' 累积分布函数 Cumulative distribution function'''：该函数评估离散随机变量X取小于或等于x的值的概率。

*'''[[Categorical distribution]]''': for discrete random variables with a finite set of values.
''' 分类分布 Categorical Distribution'''：适用于具有有限值集的离散随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】本句中“适用于具有有限值集”改为“取值结果为有限集合”。

=== ''' Functions for continuous variables 连续变量函数'''===

* '''[[Probability density function]] (pdf):''' function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.
''' 概率密度函数（pdf）Probability Density function'''：可以将其在样本空间中任意给定样本（或点）上的值（随机变量可能获得的一组值）的值解释为提供随机变量值将具有的相对可能性的函数等于那个样本。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“在样本空间（随机变量可能取值的集合）中任意给定样本（或样本点），此样本在该函数上的取值可以被解释成给出了随机变量取值等于此样本的相对可能性 relative likelihood”。

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

由此可见，除了 u0、 u1、 ... 之外，x 取任何值的概率为零，因此可以将 x 写为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句应位于下文，重复出现。

* '''Continuous probability distribution function''': most often reserved for continuous random variables.
'''连续概率分布函数 Continuous Probability Distribution Function'''：最常保留的连续随机变量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“保留的”改为“用于”。

* '''[[Cumulative distribution function]]''': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for continuous variable.
''' 累积分布函数 Cumulative distribution function'''：评估连续变量X取小于或等于x的值的概率的函数。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“评估”改为“计算”；“取小于或等于x的值”改为“取值小于或等于x”。

===Basic terms 基本术语===

* [[Mode (statistics)|'''Mode''']]: for a discrete random variable, the value with highest probability; for a continuous random variable, a location at which the probability density function has a local peak.
模式：对于离散随机变量，该值具有最高概率；对于连续随机变量，是概率密度函数具有局部峰值的位置。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“模式”改为“众数”。

* [[Support (mathematics)|'''Support''']]: set of values that can be assumed, with non-zero probability, by the random variable.
支持：可以由随机变量以非零概率假定的一组值。对于随机变量X，有时表示为R_ {X}

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“支持”改为“支撑集”；“一组值”改为“值的集合”；“有时表示为R_ {X}”改为“它的支撑集有时表示为R_ {X}”。

* '''Tail''':<ref name='tail'>More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]</ref> the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form <math>X > a</math>, <math>X < b</math> or a union thereof.
尾巴：如果pmf或pdf相对较低，则靠近随机变量边界的区域。通常形式为X> a，X <b或它们的并集。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句前一半改为“尾部：当pmf或pdf相对较低时，靠近随机变量边界的区域。”

*'''Head''':<ref name='tail' /> the region where the pmf or pdf is relatively high. Usually has the form <math>a < X < b</math>.
头部：pmf或pdf较高的区域。通常具有a <X <b的形式

* '''[[Expected value]]''' or '''mean''': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.
期望值或均值：可能值的加权平均值，以其概率作为权重；或其连续类似物。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“或其连续类似物”改为“或连续随机变量的类似取值。”

A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line. They are uniquely characterized by a cumulative density function that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

一个连续的概率分布是一个支持不可数集的概率分布，比如实数行中的一个区间。它们是唯一拥有属性的一个累积密度函数，可以用来计算支持的每个子集的概率。有许多连续概率分布的例子: 正态、均匀、卡方等等。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此段应出现在下文，此处重复。

*'''[[Median]]''': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
中位数：这样的值，即一组值小于中位数，而该组大于中位数，每一个的概率不大于二分之一。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“中位数：此值使得小于中位数的取值集合和大于中位数的取值集合各自的概率都不大于二分之一。”

*'''[[Variance]]''': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
方差：关于均值的pmf或pdf的第二矩；分布的重要指标。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“方差：关于均值的pmf或pdf的二阶矩；度量分布离散性的重要指标。”

* '''[[Standard deviation]]''': the square root of the variance, and hence another measure of dispersion.
标准偏差：方差的平方根，因此是色散的另一种度量。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“标准差：方差的平方根，因此是度量离散程度的另一指标。”

* [[Symmetric probability distribution|'''Symmetry''']]: a property of some distributions in which the portion of the distribution to the left of a specific value(usually the median) is a mirror image of the portion to its right.
对称性：某些分布的一种属性，其中特定值左侧（通常是中位数）的分布部分是其右侧部分的镜像。

*'''[[Skewness]]''': a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third [[standardized moment]] of the distribution.
偏度：衡量pmf或pdf在其均值的一侧“倾斜”的程度。分布的第三个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句“分布的第三个标准化时刻”改为“分布的三阶矩”。

*'''[[Kurtosis]]''': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
峰度：pmf或pdf尾部“脂肪”的量度。分布的第四个标准化时刻。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句改为“峰度：pmf或pdf尾部“胖瘦”的量度。分布的四阶矩。”。

==''' Discrete probability distribution 离散概率分布'''==

[[File:Discrete probability distrib.svg|right|thumb|
图4:The probability mass function of a discrete probability distribution. The probabilities of the [[Singleton (mathematics)|singleton]]s {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.]]

[[File:Discrete probability distribution.svg|right|thumb|
图5:The [[cumulative distribution function|cdf]] of a discrete probability distribution, ...离散概率分布]]

[[File:Normal probability distribution.svg|right|thumb|
图6:... of a continuous probability distribution, ...连续概率分布]]

[[File:Mixed probability distribution.svg|right|thumb|
图7:... of a distribution which has both a continuous part and a discrete part.既有连续部分又有离散部分]]

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图4图注未译，应为“离散概率分布的概率质量函数。单元集{1}, {3}, 和{7}的概率分别为0.2, 0.5, 0.3。不包含这些点的集合的概率是0。”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图5图注改为“离散概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图6图注改为“连续概率分布的累积分布函数”。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】图7图注改为“既有连续部分又有离散部分的分布的累积分布函数”。

A '''discrete probability distribution''' is a probability distribution that can take on a countable number of values.<ref>{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878591|location=New York|pages=51|oclc=710149819}}</ref> For the probabilities to add up to 1, they have to decline to zero fast enough. For example, if <math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

离散概率分布是可以具有可数数量的值的概率分布。在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1。例如，如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2，概率之和为1/2 + 1/4 + 1/8 + ... = 1。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在值的范围是无限大的情况下，这些值必须足够快地下降到零，以使概率加起来为1”一句中“无限大”改为“可数无穷大”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】含公式句中“如果<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math> for ''n'' = 1, 2”改为“如果对于''n'' = 1, 2, ...有<math>\operatorname{P}(X=n) = \tfrac{1}{2^n}</math>”。

Well-known discrete probability distributions used in statistical modeling include the [[Poisson distribution]], the [[Bernoulli distribution]], the [[binomial distribution]], the [[geometric distribution]], and the [[negative binomial distribution]].<ref name=":1" /> Additionally, the [[Uniform distribution (discrete)|discrete uniform distribution]] is commonly used in computer programs that make equal-probability random selections between a number of choices.
统计建模中使用的众所周知的离散概率分布包括泊松分布，伯努利分布，二项式分布，几何分布和负二项式分布。[3]此外，离散均匀分布通常用于在多个选择之间进行等概率随机选择的计算机程序中。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“统计建模中使用的众所周知的离散概率分布包括”中“使用的众所周知的”改为“常用的”。

When a [[Sample (statistics)|sample]] (a set of observations) is drawn from a larger population, the sample points have an [[empirical distribution function|empirical distribution]] that is discrete and that provides information about the population distribution.
当从更大的总体中抽取一个样本（一组观察值）时，这些采样点的经验分布是离散的，并且提供了有关总体分布的信息。

===Measure theoretic formulation 测量理论公式===

A [[measurable function]] <math>X \colon A \to B </math> between a [[probability space]] <math>(A, \mathcal A, P)</math> and a [[measurable space]] <math>(B, \mathcal B) </math> is called a '''discrete random variable''' provided that its image is a countable set. In this case measurability of <math>X</math> means that the pre-images of singleton sets are measurable, i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> for all <math>b \in B</math>.
The latter requirement induces a [[probability mass function]] <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. Since the pre-images of disjoint sets
are disjoint,
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
This recovers the definition given above.

一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“一个可测量的函数 <math>X \colon A \to B </math> 在一个概率空间中 <math>(A, \mathcal A, P)</math> and 和一个可测量空间 <math>(B, \mathcal B) </math> 被叫做离散随机变量。该图像是一个可数的集合。”一句改为“在概率空间<math>(A, \mathcal A, P)</math> 和可测空间之间的一个可测函数 <math>X \colon A \to B </math>称为离散随机变量，它的像是一个可数集合。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在这种情况下<math>X</math>的测量意味着单例集的原像是可测量的 i.e., <math>X^{-1}(\{b\}) \in \mathcal A</math> 对于所有的<math>b \in B</math>.”一句改为“在这种情况下<math>X</math>的可测量性意味着单元集的原像是可测量的，即对于所有的<math>b \in B</math>，有 <math>X^{-1}(\{b\}) \in \mathcal A</math> 。”

后者需要包括概率质量函数 <math>f_X \colon X(A) \to \mathbb R</math> via <math> f_X(b):=P(X^{-1}(\{b\}))</math>. 由于不相交集的原像不相交
:<math>\sum_{b \in X(A)} f_X(b) = \sum_{b \in X(A)} P(X^{-1} (\{b\})) = P \left( \bigcup_{b \in X(A)} X^{-1}(\{b\}) \right) = P(A)=1.</math>
这包含了上面所提到的定义

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“后者需要”改为“后一个必要条件”。

===''' 累积分布函数 Cumulative distribution function'''===

Equivalently to the above, a discrete random variable can be defined as a random variable whose [[cumulative distribution function]] (cdf) increases only by [[jump discontinuity|jump discontinuities]]—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. Note however that the points where the cdf jumps may form a dense set of the real numbers. The points where jumps occur are precisely the values which the random variable may take.

与上述等效，可以将离散随机变量定义为其累积分布函数（cdf）仅因跳跃不连续性而增加的随机变量，也就是说，其cdf仅在“跳跃”至较高值时才增加，并且在那些跳跃点间是常数。但是请注意，cdf跳转的点可能会形成密集的实数集。发生跳变的点恰好是随机变量可能取的值。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】此句中“密集的实数集”改为“实数的稠密集”。

===Delta-function representation 三角函数表示===

Consequently, a discrete probability distribution is often represented as a generalized [[probability density function]] involving [[Dirac delta function]]s, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.<ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004|title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology|language=en|volume=35|issue=2|pages=185–195|doi=10.1080/00207390310001638313|issn=0020-739X}}</ref>

因此，离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数，该函数实质上统一了对连续分布和离散分布的处理。当处理涉及连续和离散部分的概率分布时，这特别有用。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“三角函数表示”改为“&delta函数表示”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“离散概率分布通常表示为涉及Dirac delta函数的广义概率密度函数”改为“离散概率分布通常表示为包含狄拉克&delta函数的广义概率密度函数”。

===Indicator-function representation 指标功能表示===

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“指标功能表示”改为“指示函数表示”。
For a discrete random variable ''X'', let ''u''0, ''u''1, ... be the values it can take with non-zero probability. Denote
对于离散随机变量X，令u0，u1，...是它可以以非零概率获取的值。表示
:<math>\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“表示”改为“令”。
这些是不相交的集合，对于这样的集合：

:<math>P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.</math>

因此，X取u0，u1，...以外的任何值的概率为零，因此可以将X写入为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】标题“写入为”改为“写为”。

:<math>X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)</math>

except on a set of probability zero, where <math>1_A</math> is the [[indicator function]] of ''A''. This may serve as an alternative definition of discrete random variables.
除了概率为零的集合外，其中1_ {A}是A的指标函数。这可以用作离散随机变量的替代定义。

==Continuous probability distribution 连续概率分布==

{{See also|Probability density function}}

A '''continuous probability distribution''' is a probability distribution whose support is an uncountable set, such as an interval in the real line.<ref>{{Cite book|title=Introduction to probability models|author1=Sheldon M. Ross|date=2010|publisher=Elsevier}}</ref> They are uniquely characterized by a [[cumulative density function]]{{dn|date=August 2020}} that can be used to calculate the probability for each subset of the support. There are many examples of continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Continuous distributions|others]].
连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]。连续概率分布有很多示例：正态分布，均匀分布，卡方分布和其他分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“连续概率分布是一种概率分布，其支持是不可计数的集合，例如实线中的间隔。”一句改为“连续概率分布是支撑集为不可数集的概率分布，如实数轴上的区间。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“它们的独特之处在于可用于计算支撑的每个子集的概率的累积密度函数[需要消除歧义]”一句中，“支撑的每个子集”改为“支撑集每个子集”。

A random variable <math>X</math> has a continuous probability distribution if there is a function <math>f: \mathbb{R} \rightarrow [0, \infty)</math> such that for each interval <math>I \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> For example, if <math>I = [a, b]</math> then we would have:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

一个随机变量<math>X</math> 有一个连续的概率分布，如果这有一个函数<math>f: \mathbb{R} \rightarrow [0, \infty)</math> 对于每一个区间<math>I \subset \mathbb{R}</math> <math>I</math>的概率 <math>X</math>是<math>f</math> 在 <math>I</math>上的积分.<ref>Chapter 3.2 of {{harvp|DeGroot, Morris H.|Schervish, Mark J.|2002}}</ref> 例如, 如果<math>I = [a, b]</math> 可得到:

:<math>\operatorname{P}\left[a \le X \le b\right] = \int_a^b f(x) \, dx</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“例如”前一句中，将“如果”以后内容作为前半句，“对于”前加“使得”二字，“<math>I \subset \mathbb{R}</math>”后加逗号，“<math>I</math>的概率 <math>X</math>”改为“随机变量 <math>X</math>属于<math>I</math>的概率”，“一个随机变量<math>X</math> 有一个连续的概率分布”改为“则称这个随机变量<math>X</math> 有一个连续的概率分布”并置于后半句。

In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero. A variable that satisfies the above is called '''continuous random variable'''. Its cumulative density function is defined as

:<math>F(x) = \operatorname{P}\left[-\infty < X \le x\right] = \int_{-\infty}^x f(x) \, dx</math>

特别是，X取任何单个值a（即a≤X≤a）的概率为零，因为上下限一致的积分始终等于零。满足上述条件的变量称为连续随机变量。其累积密度函数定义为

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“特别是”改为“特别地”，“其累积密度函数定义为”下面应有上方那行公式。

which, by this definition, has the properties:
根据定义有以下一些性质

<ul>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>

<li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>\lim_{x \rightarrow -\infty} F(x) = 0</math> and <math>\lim_{x \rightarrow \infty} F(x) = 1</math>;</li>

<li style="margin: 0.7rem 0;"><math>P(a \le X < b) = F(b) - F(a)</math>; and</li>

<li style="margin: 0.7rem 0;"><math>F(x)</math> is continuous (due to the [[Riemann integral]] properties).</li>

</ul>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】以上性质没有翻译。第一行应为“<math>F(x)</math>是非减的”；第三行"and"改为“和”；第四行"and"改为“以及”；第五行应为“由于黎曼积分的性质，<math>F(x)</math>连续”

It is also possible to think in the opposite direction, which allows more flexibility. Say <math>F(x)</math> is a function that satisfies all but the last of the properties above, then <math>F</math> represents the cumulative density function for some random variable: a discrete random variable if <math>F</math> is a step function, and a continuous random variable otherwise.<ref>See Theorem 2.1 of {{harvp|Vapnik|1998}}, or [[Lebesgue's decomposition theorem]]. The section [[#Delta-function_representation]] may also be of interest.</ref> This allows for continuous distributions that has a cumulative density function, but not a probability density function, such as the [[Cantor distribution]].
也可以朝相反的方向思考，这样可以有更大的灵活性。假设 f(x)是满足上述所有性质的函数，那么 f 表示某个随机变量的累积密度函数: 如果 f 是阶跃函数，则为离散随机变量，否则为连续随机变量。这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“假设 f(x)是满足上述所有性质的函数”改为“假设 f(x)是满足上述除最后一条外其他所有性质的函数”，“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

It is often necessary to generalize the above definition for more arbitrary subsets of the real line. In these contexts, a continuous probability distribution is defined as a probability distribution with a cumulative distribution function that is [[absolute continuity|absolutely continuous]]. Equivalently, it is a probability distribution on the [[real numbers]] that is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]]. Such distributions can be represented by their [[probability density function]]s. If <math>X</math> is such an absolutely continuous random variable, then it has a [[probability density function]] <math>f(x)</math>, and its probability of falling into a Lebesgue-measurable set <math>A \subset \mathbb{R}</math> is:

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

where <math>\mu</math> is the Lebesgue measure.

对于实线的更多任意子集，通常有必要对上述定义进行概括。在这些情况下，连续概率分布定义为具有绝对连续的累积分布函数的概率分布。等效地，就Lebesgue测度而言，它是实数上的概率分布，它是绝对连续的。这样的分布可以用它们的概率密度函数表示。如果X是这样一个绝对连续的随机变量，则它具有概率密度函数f（x），并且落入Lebesgue可测量集合A⊂R的概率为：

:<math>\operatorname{P}\left[X \in A\right] = \int_A f(x) \, d\mu</math>

这里，mu是Lebesgue度量

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“对于实线的更多任意子集，通常有必要对上述定义进行概括”中“实线”改为“实数轴”。“那么 f 表示某个随机变量的累积密度函数”中“某个”改为“某种”。“这允许具有累积密度函数的连续分布，而不是概率密度函数分布，例如 Cantor 分布”改为“这允许连续分布有累积密度函数但无概率密度函数，如康托分布”。

Note on terminology: some authors use the term "continuous distribution" to denote distributions whose cumulative distribution functions are [[continuous function|continuous]], rather than [[absolute continuity|absolutely continuous]]. These distributions are the ones <math>\mu</math> such that <math>\mu\{x\}\,=\,0</math> for all <math>\,x</math>. This definition includes the (absolutely) continuous distributions defined above, but it also includes [[singular distribution]]s, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the [[Cantor distribution]].

关于术语的注释：一些作者使用术语“连续分布”来表示其累积分布函数是连续的而不是绝对连续的分布。这些分布是所有x的μ{x} = 0的μ分布。该定义包括上面定义的（绝对）连续分布，但也包括奇异分布，既不是绝对连续也不是离散的，也不是它们的混合。没有密度。 Cantor分布给出了一个示例。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“既不是绝对连续也不是离散的，也不是它们的混合。没有密度。”改为“这种奇异分布既不是绝对连续，也不是离散的，也不是它们的混合，同时也没有密度。”

== ''' [[Andrey Kolmogorov|Kolmogorov]] definition 柯尔莫哥洛夫的定义'''==

{{Main|Probability space|Probability measure}}

In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of ''X''''' is the [[pushforward measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company|year=1950|isbn=|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|url-status=live|archive-url=|archive-date=|access-date=December 5, 2019}}</ref>

在概率论的度量理论形式化中，将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）。给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P，它是满足<math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>的概率量度

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“在概率论的度量理论形式化中”一句中“度量理论”改为“测度论”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“将随机变量定义为可测量函数概率空间中的X（Ω，F，P）到一个可测量的空间（X，A）”一句改为“随机变量定义为从概率空间<math>(\Omega, \mathcal{F}, \mathbb{P})</math>到测度空间<math>(\mathcal{X},\mathcal{A})</math>的测度函数<math>(\mathcal{X},\mathcal{A})</math>。”
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“给定{ω∈Ω∣X（ω）∈A}形式的事件的概率。满足Kolmogorov的概率公理，X的概率分布为X的前推量度X * P”一句改为“假设这种形式的事件概率<math>\{\omega\in\Omega\mid X(\omega)\in A\}</math>满足满足柯尔莫哥洛夫的概率公理，则X的概率分布为X的pushforward measure<math>X_*\mathbb{P}</math>”。

==Random number generation 随机数生成==
{{Main|Pseudo-random number sampling}}

Most algorithms are based on a [[pseudorandom number generator]] that produces numbers ''X'' that are uniformly distributed in the [[half-open interval]] [0,1). These [[random variate]]s ''X'' are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last1=Dekking|first1=Frederik Michel|title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics|pages=1–11|publisher=Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|doi=10.1007/1-84628-168-7_1}}</ref>

大多数算法基于''' 伪随机数生成器 Pseudorandom Number Generator'''，该伪随机数生成器生成在半开间隔[0,1）中均匀分布的数字X。然后，通过某种算法对这些随机变量X进行转换，以创建具有所需概率分布的新随机变量。利用这种统一的伪随机源，可以生成任何随机变量的实现。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“半开间隔”改为“半开区间”。

For example, suppose <math>U</math> has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some <math>0 , we define

<math>{\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

so that

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

例如，假设U具有介于0和1之间的均匀分布。为某些对象构造一个随机的Bernoulli变量0 {\displaystyle X ={\begin{cases}1,&{\mbox{if }}U<p\\0,&{\mbox{if }}U\geq p\end{cases}}}
</math>

因此

<math>\textrm{P}(X=1) = \textrm{P}(U<p) = p,
\textrm{P}(X=0) = \textrm{P}(U\geq p) = 1-p.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“因此”改为“使得”。

This random variable X has a Bernoulli distribution with parameter <math>p</math>.<ref name=":0"/> Note that this is a transformation of discrete random variable.
该随机变量X具有参数的伯努利分布p。请注意，这是离散随机变量的变换。

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“该随机变量X具有参数的伯努利分布p”改为“该随机变量X具有以p为参数的伯努利分布”。

For a distribution function <math>F</math> of a continuous random variable, a continuous random variable must be constructed. <math>F^{inv}</math>, an inverse function of <math>F</math>, relates to the uniform variable <math>U</math>:

<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

F或连续随机变量的分布函数F，必须构造连续随机变量。 <math>F^{inv}</math>，F的反函数，涉及均匀变量U：
<math>{U\leq F(x)} = {F^{inv}(U)\leq x}.</math>

--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F或连续随机变量的分布函数F”中“F或”改为“对于”。
--[[用户:普天星相|普天星相]]([[用户讨论:普天星相|讨论]]) 【审校】“F的反函数，涉及均匀变量U”改为“F反函数<math>F^{inv}</math>与均匀变量U有以下关系：”。

For example, suppose a random variable that has an exponential distribution <math>F(x) = 1 - e^{-\lambda x}</math> must be constructed.
例如，假设必须构造一个具有指数分布<math>F(x) = 1 - e^{-\lambda x}</math> 的随机变量。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】
<math>\begin{align}
F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\ &\Leftrightarrow e^{-\lambda x } = 1-u \\&\Leftrightarrow -\lambda x = \ln(1-u) \\ &\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)
\end{align}</math>

所以<math>F^{inv}(u) = \frac{-1}{\lambda}\ln(1-u)</math> 并且如果 <math>U</math> 有一个<math>U(0,1)</math> 分布, 然后随机变量 <math>X</math> 被定义为 <math>X = F^{inv}(U) = \frac{-1}{\lambda} \ln(1-U)</math>. 这里有一个指数分布 <math>\lambda</math>.<ref name=":0" />

A frequent problem in statistical simulations (the [[Monte Carlo method]]) is the generation of [[Pseudorandomness|pseudo-random numbers]] that are distributed in a given way.
统计模拟（蒙特卡洛方法）中经常遇到的一个问题是生成以给定方式分布的伪随机数。

== Common probability distributions and their applications 共同概率分布及其应用==
{{Main list|List of probability distributions}}

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the [[Kinetic theory of gases|kinetic properties of gases]] to the [[quantum mechanical]] description of [[fundamental particles]]. For these and many other reasons, simple [[number]]s are often inadequate for describing a quantity, while probability distributions are often more appropriate.

概率分布和内蕴其中的随机变量的概念是概率论和统计学领域的基础。几乎任何可以在群体中测量的值都存在价差或可变性（例如人的身高，金属的耐用性，销售增长，交通流量等）;几乎所有测量都存在一定的固有误差；在物理学中，从气体动力学理论到基本粒子的量子力学描述都用概率论来描述。出于各种原因，简单数字通常很难表述一个量量，而用概率分布表述通常更合适。

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see [[list of probability distributions]], which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)
以下列出了一些最常见的概率分布，按与之相关的过程类型进行分组。更完整的内容请参见概率分布列表，该列表按研究对象的性质（离散，连续，多元等）进行分组。

All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a [[mixture distribution]].
下面所有的单变量分布都是单峰的。也就是说，假设值聚集在单个值周围。实际上，实际观测量可能会聚集在多个值附近，可以使用混合分布对这种量进行建模。

=== '''Linear growth (e.g. errors, offsets) 线性增长 '''===

* [[Normal distribution]] (Gaussian distribution), for a single such quantity; the most commonly used continuous distribution
''' 正态分布（高斯分布 Normal Distribution'''，适用于单个变量；是最常用的连续分布

=== ''' Exponential growth (e.g. prices, incomes, populations) 指数增长'''===

* [[Log-normal distribution]], for a single such quantity whose log is [[Normal distribution|normally]] distributed
'''对数正态分布 Log-normal Distribution'''，适用于对数为正态分布的单个变量

* [[Pareto distribution]], for a single such quantity whose log is [[Exponential distribution|exponentially]] distributed; the prototypical [[power law]] distribution
'''帕累托分布 Pareto Distribution'''，适用于指数为正态分布的单个变量；是幂律分布的原型

=== ''' Uniformly distributed quantities 数量均匀分布'''===

* [[Discrete uniform distribution]], for a finite set of values (e.g. the outcome of a fair die)
''' 离散均匀分布 Discrete Uniform Distributed'''，用于有限的一组值（例如，均匀骰子的结果）

* [[Continuous uniform distribution]], for continuously distributed values
'''连续均匀分布 Continuous Uniform Distributed'''，用于连续分布的值

=== Bernoulli trials (yes/no events, with a given probability) 伯努利试验（给定的概率的是或否事件===

* Basic distributions 基本分布:
** [[Bernoulli distribution]], for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
''' 伯努利分布 Bernoulli Distribution'''，用于描述单个伯努利试验的结果（例如成功/失败，是/否）

** [[Binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of [[Independent (statistics)|independent]] occurrences
'''二项式分布 Binomial Distribution '''，用于描述给定总数的前提下相互独立变量的成功次数（例如，成功，赞成票等）

** [[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
'''负二项分布 Negative Binomial Distribution'''，用于二项式观察，但是关注的数量是在给定成功次数之前发生的失败次数

** [[Geometric distribution]], for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the [[negative binomial distribution]]
'''几何分布 Geometric Distribution'''，用于二项式观测，但是关注的数量是首次成功之前的失败数量；是负二项分布的特殊情况

* Related to sampling schemes over a finite population 与有限人口抽样方案有关:

** [[Hypergeometric distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using [[sampling without replacement]]
''' 超几何分布 Hypergeometric Distribution'''，描述“成功”的次数（例如成功，赞成票等），给定了一定的总出现数量，使用采样而无需替换

** [[Beta-binomial distribution]], for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a [[Pólya urn model]] (in some sense, the "opposite" of [[sampling without replacement]])
''' 贝塔二项式分布 Beta-binomial Distribution'''，对于给定的总发生次数为“阳性”的次数（例如，成功，赞成票等），使用Pólyaurn模型进行采样（在某种意义上，为“替代”而不进行替换））

=== Categorical outcomes (events with ''K'' possible outcomes, with a given probability for each outcome) 分类结果（具有K个可能结果的事件，每个结果具有给定的概率）===

* [[Categorical distribution]], for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the [[Bernoulli distribution]]
针对单个分类结果的'''分类分布 Categorical Distribution'''（例如，调查中的是/否/也许）；伯努利分布的一般化

* [[Multinomial distribution]], for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the [[binomial distribution]]
给定总结果的固定数量，针对每种类别结果的数量的'''多项式分布 Multinomial Distribution'''；二项式分布的一般化

* [[Multivariate hypergeometric distribution]], similar to the [[multinomial distribution]], but using [[sampling without replacement]]; a generalization of the [[hypergeometric distribution]]
'''多元超几何分布 Multivariate Distribution'''，类似于多项式分布，但使用采样而不进行替换；超几何分布的一般化

=== Poisson process (events that occur independently with a given rate) 泊松过程（以给定速率独立发生的事件）===

* [[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
''' 泊松分布 Poisson Distribution'''，用于给定时间段内泊松型事件的发生次数

* [[Exponential distribution]], for the time before the next Poisson-type event occurs
'''指数分布 Exponential Distribution'''，在下一次泊松型事件发生之前的时间

* [[Gamma distribution]], for the time before the next k Poisson-type events occur
'''伽马分布 Gamma Distribution'''，在接下来的k个泊松型事件发生之前的时间

=== Absolute values of vectors with normally distributed components 具有正态分布分量的向量的绝对值===

* [[Rayleigh distribution]], for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
'''瑞利分布 Rayleigh Distribution'''，用于具有高斯分布正交分量的矢量幅度分布。在具有高斯实部和虚部的RF信号中发现瑞利分布。

* [[Rice distribution]], a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in [[Rician fading]] of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.
'''莱斯分布 Rice Distribution'''，是在背景信号分量稳定的情况下瑞利分布的概括。由于多径传播而在无线电信号的Rician衰落中发现，并且在非零NMR信号中出现噪声破坏的MR图像中也发现了这种情况。

=== Normally distributed quantities operated with sum of squares (for hypothesis testing) 以平方和运算的正态分布量（用于假设检验）===

* [[Chi-squared distribution]], the distribution of a sum of squared [[standard normal]] variables; useful e.g. for inference regarding the [[sample variance]] of normally distributed samples (see [[chi-squared test]])
'''卡方分布 Chi-squared Distribution'''，标准正态变量平方和的分布；有用的关于正态分布样本的样本方差的推论（请参见卡方检验）

* [[Student's t distribution]], the distribution of the ratio of a [[standard normal]] variable and the square root of a scaled [[Chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
'''学生t分布 Student‘s t Distribution'''，标准正态变量与缩放的卡方变量的平方根之比的分布；有助于推断方差未知的正态分布样本的平均值（请参阅学生的t检验）

* [[F-distribution]], the distribution of the ratio of two scaled [[Chi squared distribution|chi squared]] variables; useful e.g. for inferences that involve comparing variances or involving [[R-squared]] (the squared [[Pearson product-moment correlation coefficient|correlation coefficient]])
'''F-分布 F-Distribution'''，两个比例卡方变量的比例分布；有用的用于涉及比较方差或涉及R平方（相关系数平方）的推论

=== As a conjugate prior distributions in Bayesian inference 作为贝叶斯推断中的共轭先验分布===
{{Main|Conjugate prior}}

* [[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
'''Beta分布 Beta Distribution'''，具有单个概率（0到1之间的实数）；与伯努利分布和二项式分布共轭

* [[Gamma distribution]], for a non-negative scaling parameter; conjugate to the rate parameter of a [[Poisson distribution]] or [[exponential distribution]], the [[Precision (statistics)|precision]] (inverse [[variance]]) of a [[normal distribution]], etc.
伽玛分布，用于非负比例缩放参数；与泊松分布或指数分布的速率参数，正态分布的精度（逆方差）等共轭。

* [[Dirichlet distribution]], for a vector of probabilities that must sum to 1; conjugate to the [[categorical distribution]] and [[multinomial distribution]]; generalization of the [[beta distribution]]
'''Dirichlet分布 Dirichlet Distribution'''，对于必须为1的概率向量；与分类分布和多项式分布共轭； beta分布的一般化

*[[Wishart distribution]], for a symmetric [[non-negative definite]] matrix; conjugate to the inverse of the [[covariance matrix]] of a [[multivariate normal distribution]]; generalization of the [[gamma distribution]]<ref>{{Cite book|title=Pattern recognition and machine learning|last=Bishop, Christopher M.|date=2006|publisher=Springer|isbn=0-387-31073-8|location=New York|oclc=71008143}}</ref>
'''Wishart分布 Wishart Distribution'''，用于对称非负定矩阵；与多元正态分布的协方差矩阵的逆共轭；伽玛分布的一般化

=== Some specialized applications of probability distributions 概率分布的一些特殊应用===

* The [[cache language model]]s and other [[Statistical Language Model|statistical language models]] used in [[natural language processing]] to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
在自然语言处理中使用的高速缓存语言模型和其他统计语言模型通过概率分布来为特定单词和单词序列的出现分配概率。

* In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's [[wavefunction]] at that point (see [[Born rule]]). Therefore, the probability distribution function of the position of a particle is described by <math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>, probability that the particle's position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} in dimension one, and a similar [[triple integral]] in dimension three. This is a key principle of quantum mechanics.<ref>{{Cite book|title=Physical chemistry for the chemical sciences|last=Chang, Raymond.|publisher=|others=Thoman, John W., Jr., 1960-|year=|isbn=978-1-68015-835-9|location=[Mill Valley, California]|pages=403–406|oclc=927509011}}</ref>
在量子力学中，在给定点处找到粒子的概率密度与该点处粒子波函数大小的平方成正比（请参阅博恩法则）。因此，粒子位置的概率分布函数描述为<math>P_{a\le x\le b} (t) = \int_a^b d x\,|\Psi(x,t)|^2 </math>，粒子位置的概率x在第一个维度中的间隔为a≤x≤b，在第三个维度中的间隔类似。这是量子力学的关键原理。

* Probabilistic load flow in [[power-flow study]] explains the uncertainties of input variables as probability distribution and provide the power flow calculation also in term of probability distribution.<ref>{{Cite book|title=2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies|last1=Chen|first1=P.|last2=Chen|first2=Z.|last3=Bak-Jensen|first3=B.|date=April 2008|isbn=978-7-900714-13-8|pages=1586–1591|chapter=Probabilistic load flow: A review|doi=10.1109/drpt.2008.4523658|s2cid=18669309}}</ref>
潮流研究中的概率潮流解释了作为概率分布的输入变量的不确定性，并以概率分布的形式提供了潮流计算。

* Prediction of natural phenomena occurrences based on previous [[frequency distribution]]s such as [[tropical cyclone]]s, hail, time in between events, etc.<ref>{{Cite book|title=Statistical methods in hydrology and hydroclimatology|last=Maity, Rajib|isbn=978-981-10-8779-0|location=Singapore|oclc=1038418263|date = 2018-04-30}}</ref>
根据先前的频率分布（例如热带气旋，冰雹，事件之间的时间等）预测自然现象的发生。

==See also 另请参见==
{{Portal|Mathematics}}
*[[Copula (statistics)]]
Copula（统计数据）
* [[Empirical probability]]
经验概率
* [[Histogram]]
直方图
* [[Likelihood function]]
似然函数
* [[Kirkwood approximation]]
柯克伍德近似
* [[Moment-generating function]]
瞬时产生功能
* [[Riemann–Stieltjes integral#Application to probability theory|Riemann–Stieltjes integral application to probability theory]]
Riemann–Stieltjes积分＃在概率论中的应用
*[[Pairwise independence]]
成对独立

=== Lists 清单===

* [[List of probability distributions]]
概率分布的清单
* [[List of statistical topics]]
统计学话题的清淡
=== Probability distributions 概率分布 ===

* [[Conditional probability distribution]]
条件概率分布
* [[Joint probability distribution]]
联合概率分布
* [[Quasiprobability distribution]]
拟概率分布

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin|}}
* {{cite journal|doi=10.1016/j.ejmp.2014.05.002|pmid=25059432|title=Data distributions in magnetic resonance images: A review|url=|journal=[[Physica Medica]]|volume=30|issue=7|pages=725–741|year=2014|last1=den Dekker|first1=A. J.|last2=Sijbers|first2=J.}}
* {{cite book|last=Vapnik|first=Vladimir Naumovich|year=1998|title=Statistical Learning Theory|publisher=John Wiley and Sons}}
{{refend}}

==External links==
{{commons|Probability distribution|Probability distribution}}
*{{springer|title=Probability distribution|id=p/p074900}}
*[http://threeplusone.com/FieldGuide.pdf Field Guide to Continuous Probability Distributions], Gavin E. Crooks.