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== Definitions ==
== Definitions定义 ==
===Probability mass function===
===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''.
===Example===
===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
*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
===Assumptions and validity===
===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:{{r|Koehrsen2019}}
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).
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，（可得二项式分布），随着试验的数量趋于无穷大，'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''也是二项式分布的极限。
每次试验的成功概率除以总试验次数，（可得二项式分布），随着试验的数量趋于无穷大，'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''也是'''<font color="#ff8000"> 二项式分布Binomial distribution</font>'''的极限。
===Probability of events for a Poisson distribution'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''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 {{mvar|k}} events in an interval is given by the equation
==== Examples of probability for Poisson distributions ====
==== Examples of probability for Poisson distributions'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''概率的示例 ====
====Once in an interval events: The special case of ''λ'' = 1 and ''k'' = 0 ====
====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 {{mvar|k}} = 0 meteorite hits in the next 100 years?
=== Examples that violate the Poisson assumptions ===
=== Examples that violate the Poisson assumptions 违反泊松假设的例子===
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'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''进行建模。
== Properties ==
== Properties 性能==
=== Descriptive statistics ===
=== '''<font color="#ff8000"> 描述统计学Descriptive statistics</font>'''===
* The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.
* 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}}
* 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}}
* 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>
* 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}}
* 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''<sup>''n''</sup>。
*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'''<font color="#32CD32">“暴露”“exposure”</font>'''。）{{r|Helske2017}}
=== Median 中值===
=== Median ===
Bounds for the median (<math>\nu</math>) of the distribution are known and are sharp:
Bounds for the median (<math>\nu</math>) of the distribution are known and are sharp:
分布的中位数(< math > nu </math >)的界限是已知的，并且是锐利的:
分布的中位数(< math > nu </math >)的界限为已知，且清晰:
=== Higher moments ===
=== Higher moments 高阶矩===
* The higher [[moment (mathematics)|moments]] ''m''<sub>''k''</sub> of the Poisson distribution about the origin are [[Touchard polynomials]] in λ:
* The higher [[moment (mathematics)|moments]] ''m''<sub>''k''</sub> 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 >
< 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''<sub>''k''</sub>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]].{{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''.
=== Other properties ===
=== 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 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
* 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'''<font color="#ff8000"> 相对熵（K-L散度）[[Kullback–Leibler divergence]]</font>'''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>