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第169行： − 在概率论和统计学中，泊松分佈是以法国数学家西莫恩·德尼·泊松命名的，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发生的一定数量的事件的概率，这些事件以一个已知的常数发生，并且独立于自上一个事件以来的时间。泊松分佈还可以用来表示其他特定间隔的事件数量，如距离、面积或体积。+ 第182行：
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第193行： − 泊松分佈模型是用来模拟一个事件在一段时间或空间内发生的次数的。+ 第201行：
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第245行： − 泊松分佈可以应用于具有大量可能事件的系统，每个可能事件都是罕见的。在正确的情况下，在一个固定的时间间隔内发生的这类事件的数量是一个带有泊松分佈的随机数。+ + 第263行：
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无编辑摘要
| 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，连接线只是眼睛的向导。
| 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 = [[File:poisson cdf.svg|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。因为一个泊松分布的变量只取整数值，所以 CDF 在 k 的整数和平坦的所有其他地方都是不连续的。
| cdf _ caption = 水平轴是索引 k，表示出现的次数。因为一个泊松分布的变量只取整数值，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>
| notation = <math>\operatorname{Pois}(\lambda)</math>
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.
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，泊松分佈是以法国数学家西莫恩·德尼·泊松命名的，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发生，并且独立于与上一个事件发生的间隔时间。泊松分佈还可以用来表示其他特定间隔的事件数量，如距离、面积或体积。
== 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''.
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.
泊松分佈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:
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 > 0的泊松分佈，如果，对于 k = 0,1,2，... ，x 的概率质量函数是:
一个离散的随机变量 x 被称为具有参数 > 0的泊松分佈，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>
:<math>\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},</math>
where
where
当
* ''e'' is [[e (mathematical constant)|Euler's number]] (''e'' = 2.71828...)
* ''e'' is [[e (mathematical constant)|Euler's number]] (''e'' = 2.71828...)
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 :
正实数等于 x 的期望值和方差 < ref > 关于证明，请参阅:
正实数等于 x 的期望值和方差 < ref > 相关证明，请参阅:
[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>
[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>
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.
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.
泊松分佈可以应用于包括大量罕见的可能事件的系统。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带有泊松分佈的随机数。
===Example===
===Example===
例子：
The Poisson distribution may be useful to model events such as
The Poisson distribution may be useful to model events such as
* The number of laser photons hitting a detector in a particular time interval
* 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===
===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}}
* 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.
* 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
*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，而是在每一小的时间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 {{mvar|k}} is a Poisson random variable, and the distribution of {{mvar|k}} is a Poisson distribution.
===Probability of events for a Poisson distribution===
===Probability of events for a Poisson distribution===
泊松分布的事件概率
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
where
where
当
* <math> \lambda </math> is the average number of events per interval
* <math> \lambda </math> is the average number of events per interval