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第1,695行： − + − ==='''<font color="#ff8000">置信区间Confidence interval </font>'''===+ 第1,779行：
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第1,785行： − 在贝叶斯推断中，泊松分佈的速率参数的共轭先验是伽玛分布。让+ 第1,845行：
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第1,879行： − 泊松分佈的应用可以在很多领域找到，包括:+ − + − + − + − + − + − + − + − + − + 第1,926行：
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无编辑摘要
=== Confidence interval ===
=== Confidence interval '''<font color="#ff8000">置信区间Confidence interval </font>'''===
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 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
=== Bayesian inference ===
=== '''<font color="#ff8000"> Bayesian inference 贝叶斯推理</font>'''===
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]].{{r|Fink1976}} Let
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. Let
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. Let
在'''<font color="#ff8000"> Bayesian inference 贝叶斯推理</font>'''中，泊松分佈的速率参数的'''<font color="#ff8000"> 共轭先验Conjugate prior</font>'''是伽玛分布。让
=== Simultaneous estimation of multiple Poisson means ===
=== 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 [[Admissible decision rule|inadmissible]]. {{r|Clevenson1975}}
== Occurrence and applications ==
== Occurrence and applications 发生与应用==
{{More citations needed|date=December 2019}}
{{More citations needed|date=December 2019}}
Applications of the Poisson distribution can be found in many fields including:
Applications of the Poisson distribution can be found in many fields including:
'''<font color="#ff8000"> 泊松分佈Poisson distribution</font>'''可以应用在很多领域，包括:
* [[Telecommunication]] example: telephone calls arriving in a system.
* [[Telecommunication]] example: telephone calls arriving in a system.
*[[电信]]示例：到达系统的电话。
* [[Astronomy]] example: photons arriving at a telescope.
* [[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}}
* [[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.
* [[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.
* [[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.
* [[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=}}
* [[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.
* [[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).
* [[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 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}}
* 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}}
* 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}}
* 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.
* Internet traffic.
*互联网堵塞。
* The number of goals in sports involving two competing teams.{{r|Hornby2014}}
* 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.
* The number of deaths per year in a given age group.