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=== Confidence interval ===

=== Confidence interval '''<font color="#ff8000">置信区间Confidence interval </font>'''===

==='''<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

=== 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:

泊松分佈的应用可以在很多领域找到，包括:

'''<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.

 

* [[Astronomy]] example: photons arriving at a telescope.

* [[Astronomy]] example: photons arriving at a telescope.

 

* [[Chemistry]] example: the [[molar mass distribution]] of a [[living polymerization]].{{r|Flory1940}}

* [[Chemistry]] example: the [[molar mass distribution]] of a [[living polymerization]].{{r|Flory1940}}

 

* [[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.

 

* [[Management]] example: customers arriving at a counter or call centre.

* [[Management]] example: customers arriving at a counter or call centre.

 

* [[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.

 

* [[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=}}

 

* [[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.

 

* [[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).

 

* 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}}

 

* 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}}

 

* 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}}

 

* Internet traffic.

* Internet traffic.

 

* The number of goals in sports involving two competing teams.{{r|Hornby2014}}

* The number of goals in sports involving two competing teams.{{r|Hornby2014}}

 

* The number of deaths per year in a given age group.

* The number of deaths per year in a given age group.