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| | 审校：fairywang | | 审校：fairywang |
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| | + | 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，一定数量的事件发生的概率，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。 |
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| − | 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.
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| − | 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.
| + | 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的衰变数。 |
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| − | 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。
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| − | --[[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”
| + | == 定义 == |
| | + | ===概率分布函数=== |
| | + | 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。 |
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| − | 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.
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| − | 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.
| + | 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: |
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| − | 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。
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| − | --[[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。”
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| − | == Definitions定义 ==
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| − | ===Probability mass function概率分布函数===
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| − | The Poisson distribution is popular for modeling the ''number of times an event occurs in an interval of time or space''.
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| − | The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space.
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| − | '''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。
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| − | 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}}
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| − | 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:
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| − | 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:
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| | :<math>\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},</math> | | :<math>\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},</math> |
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| − | where
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| − | * ''e'' is [[e (mathematical constant)|Euler's number]] (''e'' = 2.71828...) | + | * ''k''是出现次数（k是出现次数（{\displaystyle k=0,1,2...}{\displaystyle k=0,1,2...}） |
| − | | + | * ''e''是欧拉数（''e'' = 2.71828...） |
| − | * ''k''! is the [[factorial]] of ''k''. | + | * !是阶乘函数。 |
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| − | The positive [[real number]] ''λ'' is equal to the [[expected value]] of ''X'' and also to its [[variance]]<ref>For the proof, see :
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| − | The positive real number λ is equal to the expected value of X and also to its variance<ref>For the proof, see :
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| − | 正实数'''λ'''等于 x 的期望值和方差相关证明，请参阅:
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| − | <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>
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| − | <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> | + | 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> |
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| − | <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>
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| | :<math>\lambda=\operatorname{E}(X)=\operatorname{Var}(X).</math> | | :<math>\lambda=\operatorname{E}(X)=\operatorname{Var}(X).</math> |
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| − | 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.
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| − | 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.
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| | 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。 | | 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。 |
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| − | | + | ===举例=== |
| − | ===Example举例=== | |
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| − | The Poisson distribution may be useful to model events such as
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| − | The Poisson distribution may be useful to model events such as
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| | 泊松分布模型可以用来模拟事件，比如 | | 泊松分布模型可以用来模拟事件，比如 |
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| − | * The number of meteorites greater than 1 meter diameter that strike Earth in a year
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| − | * The number of patients arriving in an emergency room between 10 and 11 pm
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| − | * The number of laser photons hitting a detector in a particular time interval
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| | * 一年内撞击地球的直径大于1米的陨石数量 | | * 一年内撞击地球的直径大于1米的陨石数量 |
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| | * 在特定时间间隔内撞击探测器的激光光子数 | | * 在特定时间间隔内撞击探测器的激光光子数 |
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| − | ===Assumptions and validity假设与有效条件===
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| − | The Poisson distribution is an appropriate model if the following assumptions are true:{{r|Koehrsen2019}}
| + | ===假设与有效条件=== |
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| − | The Poisson distribution is an appropriate model if the following assumptions are true:
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| | 以下假设成立时，泊松分布模型适用: | | 以下假设成立时，泊松分布模型适用: |
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| − | * {{mvar|k}} is the number of times an event occurs in an interval and {{mvar|k}} can take values 0, 1, 2, .... | + | * 事件在一个时间间隔内发生且{{mvar|k}}可以取值0，1，2，...； |
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| − | * The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
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| − | * 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.
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| − | * 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.
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| − | * 事件在一个时间间隔内发生且{mvar | k}可以取值0，1，2，...；
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| | * 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； | | * 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； |
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| | * 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。 | | * 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。 |
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| − | If these conditions are true, then {{mvar|k}} is a Poisson random variable, and the distribution of {{mvar|k}} is a Poisson distribution.
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| − | If these conditions are true, then is a Poisson random variable, and the distribution of is a Poisson distribution.
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| − | 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'''。
| + | 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。 |
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| − | | + | 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的极限。（可参考相关分布） |
| − | 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]]).
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| − | 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).
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| − | 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）
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| − | ===Probability of events for a Poisson distribution''' {泊松分布 Poisson distribution'''的事件概率===
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| − | 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
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| − | 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
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| − | 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
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| | + | ===泊松分布的事件概率=== |
| | + | 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)。<math> \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{{mvar|k}} |
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| | :<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> |
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| − | <math>P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!}</math>
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| − | < math > p (k text { events in interval }) = frac { lambda ^ k e ^ {-lambda }{ k！{/math >
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| − | where
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| − | where
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| − | * <math> \lambda </math> is the average number of events per interval | + | * <math> \lambda </math> 是每个间隔的平均事件数 |
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| − | * ''e'' is the number 2.71828... ([[e (mathematical constant)|Euler's number]]) the base of the natural logarithms
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| − | * {{mvar|k}} takes values 0, 1, 2, ...
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| − | * {{mvar|k}}! = {{mvar|k}} × ({{mvar|k}} − 1) × ({{mvar|k}} − 2) × ... × 2 × 1 is the [[factorial]] of {{mvar|k}}.
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| − | This equation is the [[probability mass function]] (PMF) for a Poisson distribution.
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| − | This equation is the probability mass function (PMF) for a Poisson distribution.
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| − | 这个方程就是概率质量函数的泊松分布。
| + | * ''e'' 数值为 2.71828... （欧拉数）自然对数的底 |
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| | + | * {{mvar|k}}取值 0, 1, 2, ... |
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| | + | * {{mvar|k}}! = {{mvar|k}} × ({{mvar|k}} − 1) × ({{mvar|k}} − 2) × ... × 2 × 1 为k的阶乘。 |
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| − | 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
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| − | 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
| + | 这个方程就是概率质量函数 (PMF)的泊松分布。 |
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| − | 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
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| − | --[[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”
| + | 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>，那么这个方程就是适用的。然后是<math> \lambda = r t </math> (以1/time 为单位的<math> r</math>) ，以及 |
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| | : <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> |
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| − | < math > p (k text { events in interval } t) = frac {(r t) ^ k e ^ {-r t }{ k！{/math >
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| − | ==== Examples of probability for Poisson distributions''' {泊松分布 Poisson distribution'''概率的示例 ====
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| − | 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.
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| − | 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.
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| − | 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。
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| − | --[[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”
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| − | Because the average event rate is one overflow flood per 100 years, ''λ'' = 1
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| − | Because the average event rate is one overflow flood per 100 years, λ = 1
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| − | 因为平均事件率是每100年发一次洪水，λ = 1
| + | ==== 泊松分布概率的示例 ==== |
| | + | 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{{mvar|k}} = 0, 1, 2, 3, 4, 5，或6次的概率，假设（其分布）适用泊松模型。 |
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| | + | 因为平均事件率是每100年发一次洪水，''λ'' = 1 |
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| | : <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> |
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| − | <math> P(k \text{ overflow floods in 100 years}) = \frac{\lambda^k e^{-\lambda}}{k!} = \frac{1^k e^{-1}}{k!}</math>
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| − | < math > p (k text { overflow in 100 years }) = frac { lambda ^ k e ^ {-lambda }}{ k! }= frac {1 ^ k e ^ {-1}{ k！{/math >
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| | : <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> |
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| − | <math> P(k = 0 \text{ overflow floods in 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{e^{-1}}{1} \approx 0.368 </math>
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| − | < math > p (k = 0 text { overflow flood in 100 years }) = frac {1 ^ 0 e ^ {-1}{0! }= frac { e ^ {-1}{1}约0.368 </math >
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| | : <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> |
| − |
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| − | <math> P(k = 1 \text{ overflow flood in 100 years}) = \frac{1^1 e^{-1}}{1!} = \frac{e^{-1}}{1} \approx 0.368 </math>
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| − |
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| − | < math > p (k = 1 text { overflow flood in 100 years }) = frac {1 ^ 1 e ^ {-1}{1! }= frac { e ^ {-1}{1}约0.368 </math >
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| | : <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 floods in 100 years}) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2} \approx 0.184 </math>
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| − |
| |
| − | < math > p (k = 2 text { overflow flood in 100 years }) = frac {1 ^ 2 e ^ {-1}{2! }= frac { e ^ {-1}{2} approx 0.184 </math >
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| − | {{col-break}}
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| − | The table below gives the probability for 0 to 6 overflow floods in a 100-year period.
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| − | The table below gives the probability for 0 to 6 overflow floods in a 100-year period.
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| | | | |
| | 下表给出了100年内0到6次洪水泛滥的概率。 | | 下表给出了100年内0到6次洪水泛滥的概率。 |
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| − | {| class="wikitable"
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| | {| class="wikitable" | | {| class="wikitable" |
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| − | { | class = “ wikitable”
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| − | ! {{mvar|k}} !! ''P''({{mvar|k}} overflow floods in 100 years)
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| − | ! !! P( overflow floods in 100 years)
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| − | !!!100年内泛滥成灾
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| − | | + | ! {{mvar|k}} !! ''P''({{mvar|k}} 100年内''k''泛滥成灾) |
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| − | | 0|| 0.368
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| − | | 0|| 0.368
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| | | 0|| 0.368 | | | 0|| 0.368 |
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| − | | 1|| 0.368
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| − | | 1|| 0.368
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| | | 1|| 0.368 | | | 1|| 0.368 |
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| − | | 2|| 0.184
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| − | | 2|| 0.184
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| | | 2|| 0.184 | | | 2|| 0.184 |
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| − | | 3|| 0.061
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| | | 3|| 0.061 | | | 3|| 0.061 |
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| − | | 3|| 0.061
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| − | | 4|| 0.015
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| − | | 4|| 0.015
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| | | 4|| 0.015 | | | 4|| 0.015 |
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| − | | 5|| 0.003
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| | | 5|| 0.003 | | | 5|| 0.003 |
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| | | 6|| 0.0005 | | | 6|| 0.0005 |
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| − | | 6|| 0.0005
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| − | | 6|| 0.0005
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| 第364行： |
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| | {{col-break}} | | {{col-break}} |
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| − | 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}}
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| − | 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.
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| | | | |
| | 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。 | | 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.
| + | 因为平均每场比赛有2.5个进球，''λ'' = 2.5。 |
| − | | |
| − | Because the average event rate is 2.5 goals per match, λ = 2.5.
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| − | | |
| − | 因为平均每场比赛有2.5个进球，λ = 2.5个。
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| − | | |
| − | | |
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| | : <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> |
| − |
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| − | <math> P(k \text{ goals in a match}) = \frac{2.5^k e^{-2.5}}{k!}</math>
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| − | < math > p (k text { goals in a match }) = frac {2.5 ^ k e ^ {-2.5}{ k！{/math >
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| | | | |
| | : <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> |
| − |
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| − | <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>
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| − | = frac {2.5 ^ 0 e ^ {-2.5}{0! }= frac { e ^ {-2.5}{1}大约0.082
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| | | | |
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| | : <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> |
| − |
| |
| − | <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>
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| − |
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| − | = frac {2.5 ^ 1 e ^ {-2.5}{1! }= frac {2.5 e ^ {-2.5}{1}约0.205
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| | | | |
| | | | |
| | : <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} \approx 0.257 </math>
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| − |
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| − | < 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 >
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| − | {{col-break}}
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| − | The table below gives the probability for 0 to 7 goals in a match.
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| − | The table below gives the probability for 0 to 7 goals in a match.
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| | 下表给出了一场比赛中0到7个进球的概率。 | | 下表给出了一场比赛中0到7个进球的概率。 |
| 第422行： |
第152行： |
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| | {| class="wikitable" | | {| class="wikitable" |
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| − | ! {{mvar|k}} !! ''P''({{mvar|k}} goals in a World Cup soccer match)
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| − | ! !! P( goals in a World Cup soccer match)
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| − | !!!P (世界杯足球赛进球)
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| − | | + | ! {{mvar|k}} !! ''P''({{mvar|k}}世界杯足球赛进球) |
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| | | 0|| 0.082 | | | 0|| 0.082 |
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| | {{col-end}} | | {{col-end}} |
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| − | ====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==== | + | ====事件唯一发生：''λ'' = 1 与 ''k'' = 0的特殊情形==== |
| − | | + | 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(''λ'' = 1 每100年撞击一次) ，而且陨石撞击的次数紧随泊松分布之后。在接下来的100年里，被陨石击中{{mvar|k}} = 0的概率是多少？ |
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| − | | |
| − | 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?
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| − | 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?
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| − | 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”
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| | : <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} \approx 0.37 </math>
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| − |
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| − | < 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 >
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| − | 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.
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| − |
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| − | 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.
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| − |
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| − | 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。
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| − | 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.
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| − |
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| − | 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.
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| | | | |
| − | 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。
| + | 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。 |
| | | | |
| | | | |
| | + | 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.
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| | | | |
| − | 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。 |
| | | | |
| − | 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。
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| | | | |
| | + | ===违反泊松假设的例子=== |
| | | | |
| | + | 每分钟抵达学生会的学生人数可能不会遵循一个泊松分布，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）。 |
| | | | |
| − | === Examples that violate the Poisson assumptions 违反泊松假设的例子===
| |
| | | | |
| | + | 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布。 |
| | | | |
| | | | |
| − | 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).
| + | 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布进行建模。 |
| | | | |
| − | 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''' 泊松分布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)。
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| − |
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| − | 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.
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| − |
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| − | 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.
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| − |
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| − | 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''' 泊松分布Poisson distribution.'''。
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| − |
| |
| − | --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”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]].
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| − |
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| − | Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a Zero-truncated Poisson distribution.
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| − |
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| − | 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''' 泊松分布Poisson distribution.'''进行建模。
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| − | --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”改为“泊松分布”
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| − | 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]].
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| − | 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.
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| | 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。 | | 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。 |
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| − | | + | == 性能== |
| − | == Properties 性能== | + | === 描述统计学=== |
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| − | === ''' 描述统计学Descriptive statistics'''=== | |
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| − | * The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.
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| | *一个泊松分布随机变量的期望值和方差均等于λ | | *一个泊松分布随机变量的期望值和方差均等于λ |
| − | | + | *当差量指数是1.，相关系数为<math>\textstyle \lambda^{-1/2}</math>。 |
| − | * The [[coefficient of variation]] is <math>\textstyle \lambda^{-1/2}</math>, while the [[index of dispersion]] is 1.{{r|Johnson2005|p=163}}
| + | *均值的绝对平均差为 |
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| − | *当[[index of dispersion]] is 1.{{r|Johnson2005|p=163}}，相关系数为<math>\textstyle \lambda^{-1/2}</math> | |
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| − | * The [[mean absolute deviation]] about the mean is{{r|Johnson2005|p=163}}
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| − | *均值的绝对平均差为{{r|Johnson2005|p=163}} | |
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| | ::<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> |
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| − | <math>\operatorname{E}[|X-\lambda|]= \frac{2 \lambda^{\lfloor\lambda\rfloor + 1} e^{-\lambda}}{\lfloor\lambda\rfloor!}.</math>
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| − | [ | x-lambda | ] = frac {2 lambda ^ { lfloor lambda rfloor + 1} e ^ {-lambda }{ lfloor lambda rfloor！} . </math >
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| − | * 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.
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| − | * 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''<sup>''n''</sup>.
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| − | * 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}}
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| | *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。 | | *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''<sup>''n''</sup>。 | | *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''<sup>''n''</sup>。 |
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| − | === Median 中值=== | + | ===中值=== |
| − | | + | 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> |
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| − | Bounds for the median (<math>\nu</math>) of the distribution are known and are [[Mathematical jargon#sharp|sharp]]:{{r|Choi1994}}
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| − | Bounds for the median (<math>\nu</math>) of the distribution are known and are sharp:
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| − | 分布的中位数(< math > nu </math >)的界限为已知，且清晰:
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| | + | === 高阶矩=== |
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| − | : <math> \lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}. </math>
| + | *泊松分布原点的高阶矩''m''<sub>''k''</sub>是同余多项式，λ中： |
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| − | <math> \lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}. </math>
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| − | 2 le nu < lambda + frac {1}{3}.数学
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| − | === Higher moments 高阶矩===
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| − | * The higher [[moment (mathematics)|moments]] ''m''<sub>''k''</sub> of the Poisson distribution about the origin are [[touchard polynomials]] in λ: | |
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| | :: <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> |
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| − | <math> m_k = \sum_{i=0}^k \lambda^i \left\{\begin{matrix} k \\ i \end{matrix}\right\},</math>
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| − | < math > m _ k = sum { i = 0} ^ k lambda ^ i left { begin { matrix } k i end { matrix } right } ，</math >
| + | :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，当泊松分布的期望值是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。 |
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| − | *泊松分布原点的高阶矩[[moment (mathematics)|moments]] ''m''<sub>''k''</sub>是同余多项式，λ中：
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| − | : 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''.
| + | 对于非中心时刻，我们定义了<math>B=k/\lambda</math>, |
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| − | 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.
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| − | 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，当泊松分布的期望值是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。
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| − | --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”改为“泊松分布”
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| − | For the non-centered moments we define <math>B=k/\lambda</math>, then{{r|Jagadeesan2017}}
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| − | For the non-centered moments we define <math>B=k/\lambda</math>, then
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| − | 对于非中心时刻，我们定义了 < math > b = k/lambda </math >
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| | ::<math> | | ::<math> |
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| − | <math>
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| − | 《数学》
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| − | E[X^k]^{1/k} \le C\cdot
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| | E[X^k]^{1/k} \le C\cdot | | E[X^k]^{1/k} \le C\cdot |
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| − | E [ x ^ k ] ^ {1/k } le c dot
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| − | \begin{cases}
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| | \begin{cases} | | \begin{cases} |
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| − | 开始{ cases }
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| − | k/B & \text{if}\quad B < e \\
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| | k/B & \text{if}\quad B < e \\ | | k/B & \text{if}\quad B < e \\ |
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| − | k/B & text { if }方 b < e
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| | k/\log B & \text{if}\quad B\ge e | | k/\log B & \text{if}\quad B\ge e |
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| − | k/\log B & \text{if}\quad B\ge e
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| − | 如果你想要一个更好的工具，你需要一个更好的工具
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| − | \end{cases}
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| | \end{cases} | | \end{cases} |
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| − | 结束{ cases }
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| | </math> | | </math> |
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| − | </math>
| + | 其中，<math>C</math>是某个大于0的绝对常数。 |
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| − | 数学
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| − | where <math>C</math> is some absolute constant greater than 0.
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| − | where <math>C</math> is some absolute constant greater than 0.
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| − | 其中，c </math > 是某个大于0的绝对常数。
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| − | === Sums of Poisson-distributed random variables 泊松分布随机变量和=== | + | ===泊松分布随机变量和=== |
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| | : 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 [[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=}} |