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{{Probability distribution
  | name       = Poisson Distribution
  | type       = mass
  | pdf_image  = [[File:poisson pmf.svg|325px]]
  | pdf_caption = The horizontal axis is the index ''k'', the number of occurrences. ''λ'' is the expected number of occurrences, which need not be an integer. 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.
  | 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.
  | notation   = <math>\operatorname{Pois}(\lambda)</math>
  | parameters = <math>\lambda\in \mathbb{R}^+</math>  (rate)
  | support    = <math>k \in \mathbb{N}_0</math>
  | pdf        = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math>
  | cdf        = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math>
(for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the [[upper incomplete gamma function]], <math>\lfloor k\rfloor</math> is the [[floor function]], and Q is the [[Regularized Gamma function|regularized gamma function]])
  | mean       = <math>\lambda</math>
  | median     = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math>
  | mode       = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math>
  | variance   = <math>\lambda</math>
  | skewness   = <math>\lambda^{-1/2}</math>
  | kurtosis   = <math>\lambda^{-1}</math>
  | entropy    = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math>
(for large <math>\lambda</math>)
<math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent-->
  | pgf        = <math>\exp[\lambda(z - 1)]</math>
  | mgf        = <math>\exp[\lambda (e^{t} - 1)]</math>
  | char       = <math>\exp[\lambda (e^{it} - 1)]</math>
  | fisher     = <math>\frac{1}{\lambda}</math>
}}

In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPA-fr|pwasɔ̃}}; in English often rendered {{IPAc-en|'|p|w|ɑː|s|ɒ|n}}), 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 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.

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.

==Basics==
The Poisson distribution is popular for modelling the ''number of times an event occurs in an interval of time or space''.

===Example===
The Poisson distribution may be useful to model events such as
* The number of meteorites greater than 1 meter diameter that strike Earth in a year
* The number of patients arriving in an emergency room between 10 and 11 pm
* The number of photons hitting a detector in a particular time interval

===Assumptions and validity===
The Poisson distribution is an appropriate model if the following assumptions are true:{{r|Koehrsen2019}}
* {{mvar|k}} is the number of times an event occurs in an interval and {{mvar|k}} can take values 0, 1, 2, ....
* The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
* The average rate at which events occur is constant.
* 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.

: Or
* The actual probability distribution is given by a [[binomial distribution]] and the number of trials is sufficiently bigger than the number of successes one is asking about (see [[Poisson distribution#Related distributions|Related distributions]]).

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===
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

:<math>P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!}</math>

where
* <math> \lambda </math> is the average number of events per interval
* ''e'' is the number 2.71828... ([[e (mathematical constant)|Euler's number]]) the base of the natural logarithms
* {{mvar|k}} takes values 0, 1, 2, ...
* {{mvar|k}}! = {{mvar|k}} × ({{mvar|k}} − 1) × ({{mvar|k}} − 2) × ... × 2 × 1 is the [[factorial]] of {{mvar|k}}.
This equation is the [[probability mass function]] (PMF) for a Poisson distribution.

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

: <math>P(k \text{ events in interval } t) = \frac{(r t)^k e^{-r t}}{k!}</math>

==== Examples of probability for Poisson distributions ====

{{col-begin|gap=5em}}
{{col-break}}
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.

Because the average event rate is one overflow flood per 100 years, ''λ'' = 1

: <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 = 0 \text{ overflow floods in 100 years}) = \frac{1^0 e^{-1}}{0!} = \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>

: <math> P(k = 2 \text{ overflow floods in 100 years}) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2} \approx 0.184 </math>

{{col-break}}
The table below gives the probability for 0 to 6 overflow floods in a 100-year period.

{| class="wikitable"
|-
! {{mvar|k}} !! ''P''({{mvar|k}} overflow floods in 100 years)
|-
| 0|| 0.368
|-
| 1|| 0.368
|-
| 2|| 0.184
|-
| 3|| 0.061
|-
| 4|| 0.015
|-
| 5|| 0.003
|-
| 6|| 0.0005
|}
{{col-end}}

{{col-begin|gap=5em}}
{{col-break}}
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}}
Because the average event rate is 2.5 goals per match, ''λ'' = 2.5.

: <math> P(k \text{ goals in a match}) = \frac{2.5^k e^{-2.5}}{k!}</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>

: <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 = 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>

{{col-break}}
The table below gives the probability for 0 to 7 goals in a match.

{| class="wikitable"
|-
! {{mvar|k}} !! ''P''({{mvar|k}} goals in a World Cup soccer match)
|-
| 0|| 0.082
|-
| 1|| 0.205
|-
| 2|| 0.257
|-
| 3|| 0.213
|-
| 4|| 0.133
|-
| 5|| 0.067
|-
| 6|| 0.028
|-
| 7|| 0.010
|}
{{col-end}}

====Once in an interval events: The special case of ''λ'' = 1 and ''k'' = 0 ====

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?

: <math> P(k = \text{0 meteorites hit in next 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{1}{e} \approx 0.37 </math>

Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining 1&nbsp;−&nbsp;0.37&nbsp;= 0.63 is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years.
In an example above, an overflow flood occurred once every 100 years (''λ''&nbsp;=&nbsp;1). The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.

In general, if an event occurs on average once per interval (''λ''&nbsp;=&nbsp;1), and the events follow a Poisson distribution, then {{nowrap|1=''P''(0 events in next interval)&nbsp;= 0.37}}. In addition, ''P''(exactly one event in next interval) = 0.37, as shown in the table for overflow floods.

=== Examples that violate the Poisson assumptions ===

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).

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.

Among patients admitted to the intensive care unit of a hospital, the number of days that the patients spend in the ICU is not Poisson distributed because the number of days cannot be zero. The distribution may be modeled using a [[Zero-truncated Poisson distribution]].

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]].

=== Poisson regression and negative binomial regression ===

[[Poisson regression]] and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0,&nbsp;1,&nbsp;2,&nbsp;...) of the number of events or occurrences in an interval.

== History ==
The distribution was first introduced by [[Siméon Denis Poisson]] (1781–1840) and published, together with his probability theory, in 1837 in his work ''Recherches sur la probabilité des jugements en matière criminelle et en matière civile''.{{r|Poisson1837|p=205-207}} The work theorized about the number of wrongful convictions in a given country by focusing on certain [[random variable]]s ''N'' that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a [[time]]-interval of given length. The result had already been given in 1711 by [[Abraham de Moivre]] in ''De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus'' .{{r|deMoivre1711|p=219}}{{r|deMoivre1718|p=14-15}}{{r|deMoivre1721|p=193}}{{r|Johnson2005|p=157}} This makes it an example of [[Stigler's law]] and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.{{r|Stigler1982|Hald1984}}

In 1860, [[Simon Newcomb]] fitted the Poisson distribution to the number of stars found in a unit of space.{{r|Newcomb1860}}
A further practical application of this distribution was made by [[Ladislaus Bortkiewicz]] in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks;{{r|vonBortkiewitsch1898|p=23-25}} this experiment introduced the Poisson distribution to the field of [[reliability engineering]].

== Definition ==
A discrete [[random variable]] ''X{{space}}'' is said to have a Poisson distribution with parameter ''λ''&nbsp;>&nbsp;0, if, for ''k''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;..., the [[probability mass function]] of ''X{{space}}'' is given by:{{r|Yates2014|p=60}}
:<math>\!f(k; \lambda)= \Pr(X = k)= \frac{\lambda^k e^{-\lambda}}{k!},</math>
where
* ''e'' is [[e (mathematical constant)|Euler's number]] (''e'' = 2.71828...)
* ''k''! is the [[factorial]] of ''k''.

The positive [[real number]] ''λ'' is equal to the [[expected value]] of ''X'' and also to its [[variance]]<ref>For the proof, see :
[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>
:<math>\lambda=\operatorname{E}(X)=\operatorname{Var}(X).</math>

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]]. How many such events will occur during a fixed time interval? Under the right circumstances, this is a random number with a Poisson distribution.

== Properties ==

=== Descriptive statistics ===
* The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.
* The [[coefficient of variation]] is <math>\textstyle \lambda^{-1/2}</math>, while the [[index of dispersion]] is 1.{{r|Johnson2005|p=163}}
* The [[mean absolute deviation]] about the mean is{{r|Johnson2005|p=163}}
::<math>\operatorname{E}[|X-\lambda|]= \frac{2 \lambda^{\lfloor\lambda\rfloor + 1} e^{-\lambda}}{\lfloor\lambda\rfloor!}.</math>
* 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&nbsp;''λ''. This is also written as [[floor function|floor]](λ). When λ is a positive integer, the modes are ''λ'' and ''λ''&nbsp;−&nbsp;1.
* All of the [[cumulant]]s of the Poisson distribution are equal to the expected value&nbsp;''λ''. The ''n''th [[factorial moment]] of the Poisson distribution is ''λ''<sup>''n''</sup>.
* 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}}

=== Median ===

Bounds for the median (<math>\nu</math>) of the distribution are known and are sharp:{{r|Choi1994}}

: <math> \lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}. </math>

=== Higher moments ===
* 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>

: 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''.

For the non-centered moments we define <math>B=k/\lambda</math>, then{{r|Jagadeesan2017}}
::<math>
E[X^k]^{1/k} \le C\cdot
\begin{cases}
k/B & \text{if}\quad B < e \\
k/\log B & \text{if}\quad B\ge e
\end{cases}
</math>
where <math>C</math> is some absolute constant greater than 0.

=== Sums of Poisson-distributed random variables ===
: If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are [[statistical independence|independent]], and <math>\lambda=\sum_{i=1}^n \lambda_i</math>, then <math>Y = \left( \sum_{i=1}^n X_i \right) \sim \operatorname{Pois}(\lambda)</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=}}

=== Other properties ===
* 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

:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>
* Bounds for the tail probabilities of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> can be derived using a [[Chernoff bound]] argument.{{r|Mitzenmacher2005|p=97-98}}
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,

:: <math> P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.</math>

* The upper tail probability can be tightened (by a factor of at least two) as follows:{{r|Short2013}}
:: <math> P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(k\mid\lambda)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(k\mid\lambda)}})}, \text{ for } x > \lambda,</math>

: where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is the directed Kullback–Leibler divergence, as described above.

* Inequalities that relate the distribution function of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> to the [[Standard normal distribution]] function <math> \Phi(x) </math> are as follows:{{r|Short2013}}

:: <math> \Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(k\mid\lambda)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k-\lambda+1)\sqrt{2\operatorname{D}_{\text{KL}}(k+1\mid\lambda)}\right), \text{ for } k > 0,</math>

: where <math>\operatorname{D}_{\text{KL}}(k\mid\lambda)</math> is again the directed Kullback–Leibler divergence.

=== Poisson races ===

Let <math>X \sim \operatorname{Pois}(\lambda)</math> and <math>Y \sim \operatorname{Pois}(\mu)</math> be independent random variables, with <math> \lambda < \mu </math>, then we have that

:<math>
 \frac{e^{-(\sqrt{\mu} -\sqrt{\lambda})^2  }}{(\lambda + \mu)^2} - \frac{e^{-(\lambda + \mu)}}{2\sqrt{\lambda \mu}} - \frac{e^{-(\lambda + \mu)}}{4\lambda \mu} \leq P(X - Y \geq 0) \leq e^{- (\sqrt{\mu} -\sqrt{\lambda})^2}
</math>

The upper bound is proved using a standard Chernoff bound.

The lower bound can be proved by noting that <math> P(X-Y\geq0\mid X+Y=i)</math> is the probability that <math>Z \geq \frac{i}{2}</math>, where <math>Z \sim \operatorname{Bin}\left(i, \frac{\lambda}{\lambda+\mu}\right)</math>, which is bounded below by <math> \frac{1}{(i+1)^2} e^{\left(-iD\left(0.5 \| \frac{\lambda}{\lambda+\mu}\right)\right)}  </math>, where <math>D</math> is [[Kullback–Leibler divergence|relative entropy]] (See the entry on [[Binomial distribution#Tail Bounds|bounds on tails of binomial distributions]] for details). Further noting that <math> X+Y \sim \operatorname{Pois}(\lambda+\mu)</math>, and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath ''et al.''.{{r|Kamath2015}}

=== Poisson Approximation ===

Assume <math>X_1\sim\operatorname{Pois}(\lambda_1), X_2\sim\operatorname{Pois}(\lambda_2), \dots, X_n\sim\operatorname{Pois}(\lambda_n)</math> where <math>\lambda_1 + \lambda_2 + \dots + \lambda_n=1</math>, then<ref>{{Cite web | url=https://newonlinecourses.science.psu.edu/stat504/node/48/ | title=1.7.7 - Relationship between the Multinomial and Poisson &#124; STAT 504}}</ref> <math>(X_1, X_2, \dots, X_n)</math> is [[Multinomial distribution|multinomially distributed]]
<math>(X_1, X_2, \dots, X_n) \sim \operatorname{Mult}(N, \lambda_1, \lambda_2, \dots, \lambda_n)</math> conditioned on <math>N = X_1 + X_2 + \dots X_n</math>.

This means{{r|Mitzenmacher2005|p=101-102}}, among other things, that for any nonnegative function <math>f(x_1,x_2,\dots,x_n)</math>,
if <math>(Y_1, Y_2, \dots, Y_n)\sim\operatorname{Mult}(m, \mathbf{p})</math> is multinomially distributed, then
:<math>
\operatorname{E}[f(Y_1, Y_2, \dots, Y_n)] \le e\sqrt{m}\operatorname{E}[f(X_1, X_2, \dots, X_n)]
</math>
where <math>(X_1, X_2, \dots, X_n)\sim\operatorname{Pois}(\mathbf{p})</math>.

The factor of <math>e\sqrt{m}</math> can be removed if <math>f</math> is further assumed to be monotonically increasing or decreasing.

== Related distributions ==
* If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the difference <math> Y = X_1 - X_2</math> follows a [[Skellam distribution]].
* If <math>X_1 \sim \mathrm{Pois}(\lambda_1)\,</math> and <math>X_2 \sim \mathrm{Pois}(\lambda_2)\,</math> are independent, then the distribution of <math>X_1</math> conditional on <math>X_1+X_2</math> is a [[binomial distribution]].
:Specifically, if <math>X_1+X_2=k</math>, then  <math>\!X_1\sim \mathrm{Binom}(k, \lambda_1/(\lambda_1+\lambda_2))</math>.
:More generally, if ''X''<sub>1</sub>, ''X''<sub>2</sub>,..., ''X''<sub>''n''</sub> are independent Poisson random variables with parameters ''λ''<sub>1</sub>, ''λ''<sub>2</sub>,..., ''λ''<sub>''n''</sub> then
:: given <math>\sum_{j=1}^n X_j=k,</math> <math>X_i \sim \mathrm{Binom}\left(k, \frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right)</math>. In fact, <math>\{X_i\} \sim \mathrm{Multinom}\left(k, \left\{\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right\}\right)</math>.
* If <math>X \sim \mathrm{Pois}(\lambda)\,</math> and the distribution of <math>Y</math>, conditional on ''X''&nbsp;=&nbsp;''k'', is a [[binomial distribution]], <math>Y \mid (X = k) \sim \mathrm{Binom}(k, p)</math>, then the distribution of Y follows a Poisson distribution <math>Y \sim \mathrm{Pois}(\lambda \cdot p)\,</math>. In fact, if <math>\{Y_i\} </math>, conditional on X = k, follows a multinomial distribution, <math>\{Y_i\} \mid (X = k) \sim \mathrm{Multinom}\left(k, p_i\right)</math>, then each <math>Y_i</math> follows an independent Poisson distribution <math>Y_i \sim \mathrm{Pois}(\lambda \cdot p_i), \rho(Y_i, Y_j) = 0</math>.
* The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the [[expected value|expected]] number of successes remains fixed — see [[#law of rare events|law of rare events]] below. Therefore, it can be used as an approximation of the binomial distribution if ''n'' is sufficiently large and ''p'' is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and ''p'' is smaller than or equal to 0.05, and an excellent approximation if ''n''&nbsp;≥&nbsp;100 and ''np''&nbsp;≤&nbsp;10.{{r|NIST2006}}
:: <math>F_\mathrm{Binomial}(k;n, p) \approx F_\mathrm{Poisson}(k;\lambda=np)\,</math>
* The Poisson distribution is a [[special case]] of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.{{r|Zhang2013|Zhang2016}} The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a [[compound Poisson distribution#Special cases|special case]] of a [[compound Poisson distribution]].
* For sufficiently large values of λ, (say λ>1000), the [[normal distribution]] with mean λ and variance λ (standard deviation <math>\sqrt{\lambda}</math>) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [[continuity correction]] is performed, i.e., if P(''X''&nbsp;≤&nbsp;''x''), where ''x'' is a non-negative integer, is replaced by P(''X''&nbsp;≤&nbsp;''x''&nbsp;+&nbsp;0.5).
:: <math>F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)\,</math>
* [[Variance-stabilizing transformation]]: If <math>X \sim \mathrm{Pois}(\lambda)\,</math>, then 
::<math>Y = 2 \sqrt{X} \approx \mathcal{N}(2\sqrt{\lambda};1)</math>,{{r|Johnson2005|p=168}}
:and
::<math>Y = \sqrt{X} \approx \mathcal{N}(\sqrt{\lambda};1/4)</math>.{{r|McCullagh1989|p=196}}
:Under this transformation, the convergence to normality (as <math>\lambda</math> increases) is far faster than the untransformed variable.{{Citation needed|date=May 2012}} Other, slightly more complicated, variance stabilizing transformations are available,{{r|Johnson2005|p=168}} one of which is [[Anscombe transform]].{{r|Anscombe1948}} See [[Data transformation (statistics)]] for more general uses of transformations.
* If for every ''t''&nbsp;>&nbsp;0 the number of arrivals in the time interval [0,&nbsp;''t''] follows the Poisson distribution with mean ''λt'', then the sequence of inter-arrival times are independent and identically distributed [[exponential distribution|exponential]] random variables having mean&nbsp;1/''λ''.{{r|Ross2010|p=317–319}}
* The [[cumulative distribution function]]s of the Poisson and [[chi-squared distribution]]s are related in the following ways:{{r|Johnson2005|p=167}}
::<math>F_\text{Poisson}(k;\lambda)  = 1-F_{\chi^2}(2\lambda;2(k+1))  \quad\quad \text{ integer } k,</math>
:and{{r|Johnson2005|p=158}}
::<math>\Pr(X=k)=F_{\chi^2}(2\lambda;2(k+1))  -F_{\chi^2}(2\lambda;2k) .
</math>

== Occurrence ==

Applications of the Poisson distribution can be found in many fields including:{{r|Rasch1963}}
* [[Telecommunication]] example: telephone calls arriving in a system.
* [[Astronomy]] example: photons arriving at a telescope.
* [[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.
* [[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.
* [[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.

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or [[space]]. Examples of events that may be modelled as a Poisson distribution include:
   <!--
   Surely there are enough examples in this list now! However, not enough with citations
   -->
* 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 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.
* 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 jumps in a stock price in a given time interval.
* Under an assumption of [[Poisson process#Homogeneous|homogeneity]], the number of times a [[web server]] is accessed per minute.
* The number of [[mutation]]s in a given stretch of [[DNA]] after a certain amount of radiation.
* The proportion of [[cells (biology)|cells]] that will be infected at a given [[multiplicity of infection]].
* The number of bacteria in a certain amount of liquid.{{r|Koyama2016}}
* The arrival of [[photons]] on a pixel circuit at a given illumination and over a given time period.
* The targeting of [[V-1 flying bomb]]s on London during World War II investigated by R. D. Clarke in 1946.{{r|Clarke1946}}
[[Patrick X. Gallagher|Gallagher]] showed in 1976 that the counts of [[prime number]]s in short intervals obey a Poisson distribution{{r|Gallagher1976}} provided a certain version of the unproved [[Second Hardy–Littlewood conjecture| prime r-tuple conjecture of Hardy-Littlewood]]{{r|Hardy1923}} is true.

{{Anchor|law of rare events}}
=== Law of rare events ===
{{main|Poisson limit theorem}}
[[File:Binomial versus poisson.svg|right|upright=1.5|thumb |Comparison of the Poisson distribution (black lines) and the [[binomial distribution]] with ''n''&nbsp;=&nbsp;10 (red circles), ''n''&nbsp;=&nbsp;20 (blue circles), ''n''&nbsp;=&nbsp;1000 (green circles). All distributions have a mean of&nbsp;5. The horizontal axis shows the number of events&nbsp;''k''. As ''n'' gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.]]

The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a [[Bernoulli trial]], where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.
In this case the binomial distribution converges to what is known as the Poisson distribution by the [[Poisson limit theorem]].

In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the [[binomial distribution]], that is

:<math>X \sim \textrm{B}(n,p). \,</math>

In such cases ''n'' is very large and ''p'' is very small (and so the expectation ''np'' is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution{{Citation needed|date=April 2012}}

:<math>X \sim \textrm{Pois}(np). \,</math>

This approximation is sometimes known as the ''law of rare events'',{{r|Cameron1998|p=5}}since each of the ''n'' individual [[Bernoulli distribution|Bernoulli events]] rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter ''np'' is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

The word ''law'' is sometimes used as a synonym of [[probability distribution]], and ''convergence in law'' means ''convergence in distribution''. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. ''The Law of Small Numbers'' is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.{{r|vonBortkiewitsch1898}}{{r|Edgeworth1913}}

=== Poisson point process ===

{{Main|Poisson point process}}

The Poisson distribution arises as the number of points of a [[Poisson point process]] located in some finite region. More specifically, if ''D'' is some region space, for example Euclidean space '''R'''<sup>''d''</sup>, for which |''D''|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if {{nowrap|''N''(''D'')}} denotes the number of points in ''D'', then

: <math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>

=== Other applications in science ===

In a Poisson process, the number of observed occurrences fluctuates about its mean ''λ'' with a [[standard deviation]] <math>\sigma_k =\sqrt{\lambda}</math>. These fluctuations are denoted as ''Poisson noise'' or (particularly in electronics) as ''[[shot noise]]''.

The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, ''even if that contribution is too small to be detected directly''. For example, the charge ''e'' on an electron can be estimated by correlating the magnitude of an [[electric current]] with its [[shot noise]]. If ''N'' electrons pass a point in a given time ''t'' on the average, the [[mean]] [[Electric current|current]] is <math>I=eN/t</math>; since the current fluctuations should be of the order <math>\sigma_I=e\sqrt{N}/t</math> (i.e., the standard deviation of the [[Poisson process]]), the charge <math>e</math> can be estimated from the ratio <math>t\sigma_I^2/I</math>.{{Citation needed|date=April 2012}}

An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced [[silver]] grains, not to the individual grains themselves. By [[Correlation|correlating]] the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).{{Citation needed|date=April 2012}} Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of [[receptor (biochemistry)|receptor]] molecules in a [[cell membrane]].
: <math>
    \Pr(N_t=k) = f(k;\lambda t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}.</math>

In [[Causal Set]] theory the discrete elements of spacetime follow a Poisson distribution in the volume.

== Generating Poisson-distributed random variables ==

A simple algorithm to generate random Poisson-distributed numbers ([[pseudo-random number sampling]]) has been given by [[Donald Knuth|Knuth]]:{{r|Knuth1997|p=137-138}}

 '''algorithm''' ''poisson random number (Knuth)'':
     '''init''':
          '''Let''' L ← ''e''<sup>−λ</sup>, k ← 0 and p ← 1.
     '''do''':
          k ← k + 1.
          Generate uniform random number u in [0,1] and '''let''' p ← p × u.
     '''while''' p > L.
     '''return''' k − 1.

The complexity is linear in the returned value ''k'', which is λ on average.  There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see {{slink||References}} below.

For large values of λ, the value of L = ''e''<sup>−λ</sup> may be so small that it is hard to represent.  This can be solved by a change to the algorithm which uses an additional parameter STEP such that ''e''<sup>−STEP</sup> does not underflow: {{Citation needed|reason=Original source is missing|date=March 2019}}

 '''algorithm''' ''poisson random number (Junhao, based on Knuth)'':
     '''init''':
          '''Let''' λLeft ← λ, k ← 0 and p ← 1.
     '''do''':
          k ← k + 1.
          Generate uniform random number u in (0,1) and '''let''' p ← p × u.
          '''while''' p < 1 and λLeft > 0:
               '''if''' λLeft > STEP:
                    p ← p × ''e''<sup>STEP</sup>
                    λLeft ← λLeft − STEP
               '''else''':
                    p ← p × ''e''<sup>λLeft</sup>
                    λLeft ← 0
     '''while''' p > 1.
     '''return''' k − 1.

The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near ''e''<sup>700</sup>, so 500 shall be a safe ''STEP''.

Other solutions for large values of λ include [[rejection sampling]] and using Gaussian approximation.

[[Inverse transform sampling]] is simple and efficient for small values of λ, and requires only one uniform random number ''u'' per sample. Cumulative probabilities are examined in turn until one exceeds ''u''.

 '''algorithm''' ''Poisson generator based upon the inversion by sequential search'':{{r|Devroye1986|p=505}}
     '''init''':
          '''Let''' x ← 0, p ← ''e''<sup>−λ</sup>, s ← p.
          Generate uniform random number u in [0,1].
     '''while''' u > s '''do''':
          x ← x + 1.
          p ← p * λ / x.
          s ← s + p.
     '''return''' x.

== Parameter estimation ==
{{See also|Poisson regression}}

=== Maximum likelihood ===
Given a sample of ''n'' measured values <math> k_i \in \{0,1,...\}</math>, for ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'', we wish to estimate the value of the parameter ''λ'' of the Poisson population from which the sample was drawn. The [[maximum likelihood]] estimate is <ref>{{cite web |last=Paszek|first=Ewa |title=Maximum Likelihood Estimation – Examples |url = http://cnx.org/content/m13500/latest/?collection=col10343/latest}}</ref>

: <math>\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i. \!</math>

Since each observation has expectation λ so does this sample mean. Therefore, the maximum likelihood estimate is an [[unbiased estimator]] of λ. It is also an efficient estimator, i.e. its estimation variance achieves the [[Cramér–Rao lower bound]] (CRLB).{{Citation needed|date=April 2012}} Hence it is [[Minimum-variance unbiased estimator|minimum-variance unbiased]]. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

To prove sufficiency we may use the [[Sufficient statistic|factorization theorem]]. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math>\mathbf{x}</math> (called <math>h(\mathbf{x})</math>) and one that depends on the parameter <math>\lambda</math> and the sample <math>\mathbf{x}</math> only through the function <math>T(\mathbf{x})</math>. Then <math>T(\mathbf{x})</math> is a sufficient statistic for <math>\lambda</math>.

: <math> P(\mathbf{x})=\prod_{i=1}^n\frac{\lambda^{x_i} e^{-\lambda}}{x_i!}=\frac{1}{\prod_{i=1}^n x_i!} \times \lambda^{\sum_{i=1}^n x_i}e^{-n\lambda} </math>

The first term, <math>h(\mathbf{x})</math>, depends only on <math>\mathbf{x}</math>. The second term, <math>g(T(\mathbf{x})|\lambda)</math>, depends on the sample only through <math>T(\mathbf{x})=\sum_{i=1}^nx_i</math>. Thus, <math>T(\mathbf{x})</math> is sufficient.

To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:

: <math> \begin{align} \ell(\lambda) & = \ln \prod_{i=1}^n f(k_i \mid \lambda) \\ & = \sum_{i=1}^n \ln\!\left(\frac{e^{-\lambda}\lambda^{k_i}}{k_i!}\right) \\ & = -n\lambda + \left(\sum_{i=1}^n k_i\right) \ln(\lambda) - \sum_{i=1}^n \ln(k_i!). \end{align} </math>

We take the derivative of ''<math>\ell</math>'' with respect to ''λ'' and compare it to zero:

: <math>\frac{\mathrm{d}}{\mathrm{d}\lambda} \ell(\lambda) = 0 \iff -n + \left(\sum_{i=1}^n k_i\right) \frac{1}{\lambda} = 0. \!</math>

Solving for ''λ'' gives a stationary point.

: <math> \lambda = \frac{\sum_{i=1}^n k_i}{n}</math>

So ''λ'' is the average of the ''k''<sub>''i''</sub> values. Obtaining the sign of the second derivative of ''L'' at the stationary point will determine what kind of extreme value ''λ'' is.

: <math>\frac{\partial^2 \ell}{\partial \lambda^2} = -\lambda^{-2}\sum_{i=1}^n k_i </math>

Evaluating the second derivative ''at the stationary point'' gives:

: <math>\frac{\partial^2 \ell}{\partial \lambda^2} = - \frac{n^2}{\sum_{i=1}^n k_i} </math>

which is the negative of ''n'' times the reciprocal of the average of the k<sub>i</sub>. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.

For [[Completeness (statistics)|completeness]], a family of distributions is said to be complete if and only if <math> E(g(T)) = 0</math>  implies that <math>P_\lambda(g(T) = 0) = 1</math> for all <math>\lambda</math>. If the individual <math>X_i</math> are iid <math>\mathrm{Po}(\lambda)</math>, then <math>T(\mathbf{x})=\sum_{i=1}^nX_i\sim \mathrm{Po}(n\lambda)</math>. Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.

: <math>E(g(T))=\sum_{t=0}^\infty g(t)\frac{(n\lambda)^te^{-n\lambda}}{t!}=0</math>

For this equality to hold, <math>g(t)</math> must be 0. This follows from the fact that none of the other terms will be 0 for all <math>t</math> in the sum and for all possible values of <math>\lambda</math>. Hence, <math> E(g(T)) = 0</math> for all <math>\lambda</math> implies that <math>P_\lambda(g(T) = 0) = 1</math>, and the statistic has been shown to be complete.

=== Confidence interval ===
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

:<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>

or equivalently,

:<math>F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),</math>

where <math>\chi^{2}(p;n)</math> is the [[quantile function]] (corresponding to a lower tail area ''p'') of the chi-squared distribution with ''n'' degrees of freedom and <math>F^{-1}(p;n,1)</math> is the quantile function of a [[Gamma distribution]] with shape parameter n and scale parameter 1.{{r|Johnson2005|p=176-178|Garwood1936}} This interval is '[[exact statistics|exact]]' in the sense that its [[coverage probability]] is never less than the nominal {{math|1 – α}}.

When quantiles of the Gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the [[Chi-square distribution#Asymptotic properties|Wilson–Hilferty transformation]]):{{r|Breslow1987}}
:<math>k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3, </math>
where <math>z_{\alpha/2}</math> denotes the [[standard normal deviate]] with upper tail area {{math|α / 2}}.

For application of these formulae in the same context as above (given a sample of ''n'' measured values ''k''<sub>''i''</sub> each drawn from a Poisson distribution with mean ''λ''), one would set

:<math>k=\sum_{i=1}^n k_i ,\!</math>
calculate an interval for ''μ''&nbsp;=&nbsp;''nλ'', and then derive the interval for ''λ''.

=== Bayesian inference ===
In [[Bayesian inference]], the [[conjugate prior]] for the rate parameter ''λ'' of the Poisson distribution is the [[gamma distribution]].{{r|Fink1976}} Let

:<math>\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!</math>

denote that ''λ'' is distributed according to the gamma [[probability density function|density]] ''g'' parameterized in terms of a [[shape parameter]] ''α'' and an inverse [[scale parameter]] ''β'':

:<math> g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.</math>

Then, given the same sample of ''n'' measured values ''k''<sub>''i''</sub> [[Poisson distribution#Maximum likelihood|as before]], and a prior of Gamma(''α'', ''β''), the posterior distribution is

:<math>\lambda \sim \mathrm{Gamma}\left(\alpha + \sum_{i=1}^n k_i, \beta + n\right). \!</math>

The posterior mean E[''λ''] approaches the maximum likelihood estimate <math>\widehat{\lambda}_\mathrm{MLE}</math> in the limit as <math>\alpha\to 0,\ \beta\to 0</math>, which follows immediately from the general expression of the mean of the [[gamma distribution]].

The [[posterior predictive distribution]] for a single additional observation is a [[negative binomial distribution]],{{r|Gelman2003|p=53}} sometimes called a Gamma–Poisson distribution.

=== Simultaneous estimation of multiple Poisson means ===
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}}

In this case, a family of [[minimax estimator]]s is given for any <math>0 < c \leq 2(p-1)</math> and <math>b \geq (p-2+p^{-1})</math> as{{r|Berger1985}}
:<math>{\hat \lambda}_i = \left(1 - \frac{c}{b + \sum_{i=1}^p X_i}\right) X_i, \qquad i=1,\dots,p.</math>

== Bivariate Poisson distribution ==
This distribution has been extended to the bivariate case.{{r|Loukas1986}} The [[generating function]] for this distribution is
: <math> g( u, v ) = \exp[ ( \theta_1 - \theta_{12} )( u - 1 ) + ( \theta_2 - \theta_{12} )(v - 1) + \theta_{12} ( uv - 1 ) ] </math>

with
: <math> \theta_1, \theta_2 > \theta_{ 12 } > 0 \, </math>

The marginal distributions are Poisson(''θ''<sub>1</sub>) and Poisson(''θ''<sub>2</sub>) and the correlation coefficient is limited to the range
: <math> 0 \le \rho \le \min\left\{ \frac{ \theta_1 }{ \theta_2 }, \frac{ \theta_2 }{ \theta_1 } \right\}</math>

A simple way to generate a bivariate Poisson distribution <math>X_1,X_2</math> is to take three independent Poisson distributions <math>Y_1,Y_2,Y_3</math> with means <math>\lambda_1,\lambda_2,\lambda_3</math> and then set <math>X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3</math>. The probability function of the bivariate Poisson distribution is
: <math>
\begin{align}
& \Pr(X_1=k_1,X_2=k_2) \\
= {} & \exp\left(-\lambda_1-\lambda_2-\lambda_3\right) \frac{\lambda_1^{k_1}}{k_1!} \frac{\lambda_2^{k_2}}{k_2!} \sum_{k=0}^{\min(k_1,k_2)} \binom{k_1}{k} \binom{k_2}{k} k! \left( \frac{\lambda_3}{\lambda_1\lambda_2}\right)^k
\end{align}
</math>

==Computer software for the Poisson distribution==

The Poisson distribution poses two different tasks for dedicated software libraries: ''Evaluating'' the distribution <math>P(k;\lambda)</math>, and ''drawing random numbers'' according to that distribution.

=== Evaluating the Poisson distribution ===
Computing <math>P(k;\lambda)</math> for given <math>k</math> and <math>\lambda</math> is a trivial task that can be accomplished by using the standard definition of <math>P(k;\lambda)</math> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λ<sup>''k''</sup> and ''k''!. The fraction of λ<sup>''k''</sup> to ''k''! can also produce a rounding error that is very large compared to ''e''<sup>−λ</sup>, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as
:<math>\!f(k; \lambda)= \exp \left[ k\ln \lambda  - \lambda  - \ln \Gamma (k+1) \right],</math>
which is mathematically equivalent but numerically stable. The natural logarithm of the [[Gamma function]] can be obtained using the <code>lgamma</code> function in the [[C (programming language)]] standard library (C99 version) or [[R (programming language)]], the <code>gammaln</code> function in [[MATLAB]] or [[SciPy]], or the <code>log_gamma</code> function in [[Fortran]] 2008 and later.

Some computing languages provide built-in functions to evaluate the Poisson distribution, namely
* [[R (programming language)|R]]: function <code>dpois(x, lambda)</code>;
* [[Microsoft Excel|Excel]]: function <code>POISSON( x, mean, cumulative)</code>, with a flag to specify the cumulative distribution;
* [[Mathematica]]: univariate Poisson distribution as <code>PoissonDistribution[<math>\lambda</math>]</code>,<ref name="WLPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/PoissonDistribution.html |title = Wolfram Language: PoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref> bivariate Poisson distribution as <code>MultivariatePoissonDistribution[<math>\theta_{12}</math>,{ <math>\theta_1 - \theta_{12}</math>, <math>\theta_2 - \theta_{12}</math>}]</code>,.<ref name="WLMvPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html |title = Wolfram Language: MultivariatePoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref>

=== Random drawing from the Poisson distribution ===
The less trivial task is to draw random integers from the Poisson distribution with given <math>\lambda</math>.

Solutions are provided by:
* [[R (programming language)|R]]: function <code>rpois(n, lambda)</code>;
* [[GNU Scientific Library]] (GSL): function [https://www.gnu.org/software/gsl/doc/html/randist.html#the-poisson-distribution gsl_ran_poisson]

== See also ==
{{div col |colwidth = 18em }}
* [[Compound Poisson distribution]]
* [[Conway–Maxwell–Poisson distribution]]
* [[Erlang distribution]]
* [[Hermite distribution]]
* [[Index of dispersion]]
* [[Negative binomial distribution]]
* [[Poisson clumping]]
* [[Poisson point process]]
* [[Poisson regression]]
* [[Poisson sampling]]
* [[Poisson wavelet]]
* [[Queueing theory]]
* [[Renewal theory]]
* [[Robbins lemma]]
* [[Skellam distribution]]
* [[Tweedie distribution]]
* [[Zero-inflated model]]
* [[Zero-truncated Poisson distribution]]
{{div col end}}

== References ==
=== Citations ===
{{Reflist
|refs =
<ref name=Haight1967>
{{citation
 |last1=Haight
 |first1=Frank A.
 |title=Handbook of the Poisson Distribution
 |publisher=John Wiley & Sons
 |location=New York, NY, USA
 |year=1967
|isbn=978-0-471-33932-8
}}</ref>
<ref name=Brooks2007>
{{citation
 |last=Brooks |first=E. Bruce
 |title=Statistics &#124; The Poisson Distribution 
 |series=Warring States Project |publisher=Umass.edu 
 |date=2007-08-24 
 |accessdate=2014-04-18
 |url=https://www.umass.edu/wsp/resources/reference/poisson/
}}</ref>
<ref name=Poisson1837>
{{citation
 |last1=Poisson
 |first1=Siméon D.
 |title=Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés
 |trans-title=Research on the Probability of Judgments in Criminal and Civil Matters
 |language=French
 |publisher=Bachelier
 |location=Paris, France
 |year=1837
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k110193z/f218.image
}}</ref>
<ref name=Koehrsen2019>
{{citation
 |last1=Koehrsen
 |first1=William
 |title=The Poisson Distribution and Poisson Process Explained
 |publisher=Towards Data Science
 |date=2019-01-20
 |url=https://towardsdatascience.com/the-poisson-distribution-and-poisson-process-explained-4e2cb17d459
 |accessdate=2019-09-19
}}</ref>
<ref name="Ugarte2016">
{{Citation
 |last1=Ugarte |first1=Maria Dolores
 |last2=Militino |first2=Ana F.
 |last3=Arnholt |first3=Alan T.
 |title=Probability and Statistics with R
 |edition=Second
 |year=2016
 |publisher=CRC Press
 |location=Boca Raton, FL, USA
 |isbn=978-1-4665-0439-4
}}
</ref>
<ref name=deMoivre1711>
{{citation
 |last1=de Moivre |first1=Abraham
 |title=De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus
 |trans-title=On the Measurement of Chance, or, on the Probability of Events in Games Depending Upon Fortuitous Chance
 |language=Latin
 |journal=[[Philosophical Transactions of the Royal Society]]
 |year=1711 |volume=27 |issue=329 |page=213–264
 |doi=10.1098/rstl.1710.0018
 |doi-access=free
}}</ref>
<ref name=deMoivre1718>
{{citation
 |last1=de Moivre |first1=Abraham
 |year=1718
 |title=The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play
 |publisher=W. Pearson
 |location=London, Great Britain
 |url=https://books.google.at/books?id=3EPac6QpbuMC&pg=PA14
}}</ref>
<ref name=deMoivre1721>
{{citation
 |last1=de Moivre |first1=Abraham
 |year=1721
 |chapter=Of the Laws of Chance
 |title=The Philosophical Transactions from the Year MDCC (where Mr. Lowthorp Ends) to the Year MDCCXX. Abridg'd, and Dispos'd Under General Heads
 |volume=Vol. I
 |language=Latin
 |editor-last1=Motte |editor-first1=Benjamin
 |publisher=R. Wilkin, R. Robinson, S. Ballard, W. and J. Innys, and J. Osborn
 |location=London, Great Britain
 |page=190–219
}}</ref>
<ref name="Johnson2005">
{{citation
 |last1=Johnson |first1=Norman L.
 |last2=Kemp |first2=Adrienne W.
 |last3=Kotz |first3=Samuel
 |chapter=Poisson Distribution
 |year=2005 |pages=156-207
 |title=Univariate Discrete Distributions
 |edition=3rd
 |publisher=John Wiley & Sons, Inc. 
 |location=New York, NY, USA
 |isbn=978-0-471-27246-5
 |doi=10.1002/0471715816
}}</ref>
<ref name=Helske2017>
{{cite arxiv
 |last=Helske |first=Jouni
 |date=2017
 |title=KFAS: Exponential family state space models in R
 |eprint=1612.01907
 |class=stat.CO
}}</ref>
<ref name=Stigler1982>
{{citation
 |last1=Stigler |first1=Stephen M. 
 |title=Poisson on the Poisson Distribution
 |journal=Statistics & Probability Letters
 |year=1982 |volume=1 |issue=1 |pages=33–35
 |doi=10.1016/0167-7152(82)90010-4
}}</ref>
<ref name=Hald1984>
{{Citation
 |last1=Hald |first1=Anders
 |last2=de Moivre |first2=Abraham
 |last3=McClintock |first3=Bruce
 |title=A. de Moivre: 'De Mensura Sortis' or 'On the Measurement of Chance'
 |journal=International Statistical Review / Revue Internationale de Statistique
 |year=1984 |volume=52 |issue=3 |pages=229–262
 |doi=10.2307/1403045
 |jstor=1403045
}}</ref>
<ref name=Newcomb1860>
{{citation
 |last1=Newcomb
 |first1=Simon
 |year=1860
 |title=Notes on the theory of probabilities
 |journal=The Mathematical Monthly
 |volume=2
 |issue=4
 |pages=134–140
 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075590&seq=150
}}</ref>
<ref name=vonBortkiewitsch1898>
{{citation
 |last1=von Bortkiewitsch |first1=Ladislaus <!-- This is the spelling of the name as it appears in the book, the Polish version would be Vladislav Bortkevič--> 
 |title=Das Gesetz der kleinen Zahlen
 |trans-title=The law of small numbers
 |language=German
 |publisher=B.&nbsp;G.&nbsp;Teubner |location=Leipzig, Germany
 |year=1898 
 |page=On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/13 page 1], Bortkiewicz presents the Poisson distribution. On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/35 pages 23–25], Bortkiewitsch presents his analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.)
}}</ref>
<ref name=Lehmann1986>
{{citation
 |last1=Lehmann |first1=Erich Leo
 |title=Testing Statistical Hypotheses
 |publisher=Springer Verlag |location=New York, NJ, USA
 |edition=second
 |year=1986
 |isbn=978-0-387-94919-2}}</ref>
<ref name=Yates2014>
{{citation
 |last1=Yates |first1=Roy D. |last2=Goodman |first2=David J.
 |title=Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers
 |year=2014
 |publisher=Wiley
 |location=Hoboken, USA
 |edition=2nd |isbn=978-0-471-45259-1}}
</ref>
<ref name=Choi1994>
{{citation
 |last1=Choi |first1=Kwok P.
 |title=On the medians of gamma distributions and an equation of Ramanujan
 |journal=Proceedings of the American Mathematical Society
 |year=1994 |volume=121 |issue=1 |pages=245–251
 |doi=10.2307/2160389
 |jstor=2160389}}</ref>
<ref name=Riordan1937>
{{citation
 |last=Riordan | first=John
 |journal=Annals of Mathematical Statistics
 |title=Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions
 |year=1937 |volume=8 |issue=2 |pages=103–111
 |jstor=2957598
 |doi=10.1214/aoms/1177732430
 |url=https://projecteuclid.org/download/pdf_1/euclid.aoms/1177732430
 |format=PDF
}}</ref>
<ref name=Jagadeesan2017>
{{cite arxiv
 |last1=Jagadeesan |first1=Meena
 |title=Simple analysis of sparse, sign-consistent JL 
 |year=2017 
 |eprint=1708.02966 |class=cs.DS 
}}</ref>
<ref name=Raikov1937>
{{citation
 |last=Raikov |first=Dmitry
 |title=On the decomposition of Poisson laws
 |journal=Comptes Rendus de l'Académie des Sciences de l'URSS
 |year=1937 |volume=14 |page=9–11
}}</ref>
<ref name=vonMises1964>
{{citation
 |last=von Mises |first=Richard
 |title=Mathematical Theory of Probability and Statistics
 |year=1964
 |publisher=Academic Press
 |location=New York, NJ, USA
 |isbn=978-1-4832-3213-3
 |doi=10.1016/C2013-0-12460-9
}}</ref>
<ref name=Kamath2015>
{{ citation
 |last1=Kamath |first1=Govinda M.
 |last2=Şaşoğlu |first2=Eren
 |last3=Tse |first3=David
 |contribution=Optimal Haplotype Assembly from High-Throughput Mate-Pair Reads
 |title=2015 IEEE International Symposium on Information Theory (ISIT), 14-19 June
 |year=2015
 |pages=914–918
 |doi=10.1109/ISIT.2015.7282588
 |location=Hong Kong, China
 |arxiv=1502.01975
}}</ref>
<ref name=Laha1979>
{{citation
 |last1=Laha |first1=Radha G.
 |last2=Rohatgi |first2=Vijay K.
 |title=Probability Theory
 |publisher=John Wiley & Sons
 |location=New York, NJ, USA
 |isbn=978-0-471-03262-5
 |year=1979
}}</ref>
<ref name=Mitzenmacher2005>
{{citation
 |last1=Mitzenmacher |first1=Michael |author-link1=Michael Mitzenmacher
 |last2=Upfal |first2=Eli |author-link2=Eli Upfal
 |title=Probability and Computing: Randomized Algorithms and Probabilistic Analysis
 |publisher=Cambridge University Press 
 |location=Cambridge, UK
 |isbn=978-0-521-83540-4
 |year=2005
}}</ref>
<ref name=Short2013>
{{citation 
 |last1=Short |first1=Michael 
 |title=Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions 
 |journal=ISRN Probability and Statistics 
 |year=2013 |volume=2013 |page=412958 
 |doi=10.1155/2013/412958 |doi-access=free
}}</ref>
<ref name=NIST2006>
{{citation
 |last1=Prins |first1=Jack
 |chapter=6.3.3.1. Counts Control Charts
 |title=e-Handbook of Statistical Methods
 |chapter-url=http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm
 |date=2012
 |publisher=NIST/SEMATECH
 |accessdate=2019-09-20
}}</ref>
<ref name=Zhang2013>
{{citation
 |last1=Zhang |first1=Huiming
 |last2=Liu |first2=Yunxiao
 |last3=Li |first3=Bo 
 |title=Notes on discrete compound Poisson model with applications to risk theory
 |journal=Insurance: Mathematics and Economics 
 |year=2014 |volume=59 |pages=325–336
 |doi=10.1016/j.insmatheco.2014.09.012
}}</ref>
<ref name=Zhang2016>
{{citation
 |last1=Zhang |first1=Huiming
 |last2=Li |first2=Bo 
 |title=Characterizations of discrete compound Poisson distributions
 |journal=Communications in Statistics - Theory and Methods
 |year=2016 |volume=45 |issue=22 |pages=6789–6802
 |doi=10.1080/03610926.2014.901375
}}</ref>
<ref name=McCullagh1989>
{{citation
 |last1=McCullagh |first1=Peter |authorlink1=Peter McCullagh
 |last2=Nelder |first2=John |authorlink2=John Nelder
 |title = Generalized Linear Models
 |publisher=Chapman and Hall
 |location=London, UK
 |series=Monographs on Statistics and Applied Probability
 |year=1989
 |volume=37
 |isbn=978-0-412-31760-6
}}</ref>
<ref name=Anscombe1948>
{{citation
 |last=Anscombe |first=Francis J. |author-link=Frank Anscombe
 |title=The transformation of Poisson, binomial and negative binomial data
 |journal=Biometrika
 |year=1948 |volume=35 |issue=3-4 |pages=246–254
 |jstor=2332343 
 |doi=10.1093/biomet/35.3-4.246
}}</ref>
<ref name=Ross2010>
{{citation
 |last1=Ross |first1=Sheldon M.
 |title=Introduction to Probability Models
 |edition=tenth
 |publisher=Academic Press
 |location=Boston, MA, USA
 |year=2010
 |isbn=978-0-12-375686-2
}}</ref>
<ref name=Rasch1963>
{{Citation 
 |last1=Rasch |first1=Georg
 |contribution=The Poisson Process as a Model for a Diversity of Behavioural Phenomena
 |title=17th International Congress of Psychology
 |publisher=American Psychological Association
 |place=Washington, DC, USA, August 20th - 26th, 1963
 |year=1963
 |series=
 |volume=2 |issue= |pages=
 |doi=10.1037/e685262012-108 <!-- doi leads to a paywall -->
 |chapter-url=http://www.rasch.org/memo1963.pdf
}}</ref>
<ref name=Flory1940>
{{citation
 |last1=Flory
 |first1=Paul J.
 |title=Molecular Size Distribution in Ethylene Oxide Polymers
 |journal=Journal of the American Chemical Society
 |year=1940 |volume=62 |issue=6
 |pages=1561–1565
 |doi=10.1021/ja01863a066
}}</ref>
<ref name=Lomnitz1994>
{{citation
 |last1=Lomnitz |first1=Cinna
 |title=Fundamentals of Earthquake Prediction
 |year=1994
 |publisher=John Wiley & Sons
 |location=New York
 |oclc=647404423
 |isbn=0-471-57419-8
}}</ref>
<ref name=Student1907>
{{citation
 |last=Student |first=
 |title=On the Error of Counting with a Haemacytometer
 |journal=Biometrika
 |year=1907 |volume=5 |issue=3 |pages=351-360
 |jstor=2331633 
 |doi=10.2307/2331633
}}</ref>
<ref name=Boland1984>
{{citation
 |last=Boland |first=Philip J.
 |title=A Biographical Glimpse of William Sealy Gosset
 |journal=The American Statistician
 |year=1984
 |volume=38
 |issue=3
 |pages=179–183
 |jstor=2683648 
 |doi=10.1080/00031305.1984.10483195
}}</ref>
<ref name=Erlang1909>
{{citation
 |last1=Erlang |first1=Agner K.
 |title=Sandsynlighedsregning og Telefonsamtaler
 |trans-title=Probability Calculation and Telephone Conversations
 |language=Danish
 |journal=Nyt Tidsskrift for Matematik
 |year=1909 |volume=20 |issue=B |pages=33-39 
 |jstor=24528622
}}</ref>
<ref name=Hornby2014>
{{citation
 |last=Hornby |first=Dave
 |title=Football Prediction Model: Poisson Distribution
 |publisher=Sports Betting Online
 |url=http://www.sportsbettingonline.net/strategy/football-prediction-model-poisson-distribution
 |year=2014 
 |accessdate=2014-09-19
}}</ref>
<ref name=Koyama2016>
{{Citation 
 |last1=Koyama   |first1=Kento
 |last2=Hokunan  |first2=Hidekazu
 |last3=Hasegawa |first3=Mayumi
 |last4=Kawamura |first4=Shuso
 |last5=Koseki   |first5=Shigenobu
 |title=Do bacterial cell numbers follow a theoretical Poisson distribution? Comparison of experimentally obtained numbers of single cells with random number generation via computer simulation
 |journal=Food Microbiology
 |year=2016 |volume=60 |pages=49–53
 |doi=10.1016/j.fm.2016.05.019
}}</ref>
<ref name=Clarke1946>
{{citation
 |last1=Clarke |first1=R. D. 
 |title=An application of the Poisson distribution
 |journal=Journal of the Institute of Actuaries
 |year=1946 |volume=72 |issue=3 |page=481
 |doi=10.1017/S0020268100035435
 |url=https://www.actuaries.org.uk/system/files/documents/pdf/0481.pdf
 |doi-access=free
}}</ref>
<ref name=Gallagher1976>
{{Citation 
 |last=Gallagher |first=Patrick X.
 |title=On the distribution of primes in short intervals
 |journal=Mathematika
 |year=1976 |volume=23 |issue=1 |pages=4–9 
 |doi=10.1112/s0025579300016442
 |doi-access=free
}}</ref>
<ref name=Hardy1923>
{{Citation 
 |last1=Hardy |first1=Godfrey H. |author-link1=G. H. Hardy 
 |last2=Littlewood |first2=John E. |author-link2=John Edensor Littlewood
 |title=On some problems of "partitio numerorum" III: On the expression of a number as a sum of primes
 |journal=Acta Mathematica
 |year=1923 |volume=44 |pages=1–70 
 |doi=10.1007/BF02403921
}}</ref>
<ref name=Cameron1998>
{{citation
 |last1=Cameron |first1=A. Colin 
 |last2=Trivedi |first2=Pravin K. 
 |title=Regression Analysis of Count Data
 |year=1998
 |publisher=Cambridge University Press
 |location=Cambridge, UK
 |isbn=978-0-521-63567-7
 |url=https://books.google.com/?id=SKUXe_PjtRMC&pg=PA5
}}</ref>
<ref name=Edgeworth1913>
{{Citation
 |last=Edgeworth |first=Francis Y.
 |authorlink=Francis Ysidro Edgeworth
 |title=On the use of the theory of probabilities in statistics relating to society 
 |journal=[[Journal of the Royal Statistical Society]]
 |year=1913 |volume=76 |issue=2 |pages=165–193
 |jstor=2340091 |doi=10.2307/2340091
}}</ref>
<ref name=Knuth1997>
{{citation
 |last=Knuth |first=Donald Ervin
 |title = Seminumerical Algorithms
 |publisher=[[Addison Wesley]]
 |series=[[The Art of Computer Programming]]
 |volume=2
 |edition=3rd
 |year=1997
 |isbn=978-0-201-89684-8
}}</ref>
<ref name="Devroye1986">
{{citation
 |last=Devroye |first=Luc |author-link=Luc Devroye
 |chapter=Discrete Univariate Distributions
 |chapterurl=http://luc.devroye.org/chapter_ten.pdf
 |title=Non-Uniform Random Variate Generation
 |year=1986
 |publisher=Springer-Verlag
 |place=New York, NJ, USA
 |pages=485-553
 |url=http://luc.devroye.org/rnbookindex.html
 |doi=10.1007/978-1-4613-8643-8_10
}}</ref>
<ref name="Garwood1936">
{{citation
 |last=Garwood |first=Frank
 |title=Fiducial Limits for the Poisson Distribution
 |journal=Biometrika
 |year=1936 |volume=28 |issue=3/4 |pages=437–442
 |doi=10.1093/biomet/28.3-4.437
 |jstor=2333958
}}</ref> <ref name=Breslow1987>
{{citation 
 |last1=Breslow |first1=Norman E. |authorlink1=Norman Breslow
 |last2=Day |first2=Nick E. |authorlink2=Nick Day
 |title=Statistical Methods in Cancer Research: Volume 2—The Design and Analysis of Cohort Studies
 |year=1987
 |publisher=[[International Agency for Research on Cancer]]
 |location=Lyon, France
 |url=http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php
 |isbn=978-92-832-0182-3
}}</ref>
<ref name=Fink1976>
{{citation
 |last=Fink |first=Daniel
 |title=A Compendium of Conjugate Priors
 |year=1997
 |publisher=
 |location= 
 |isbn= 
}}</ref>
<ref name=Gelman2003>
{{citation
 |last1=Gelman |first1=
 |last2=Carlin |first2=John B.
 |last3=Stern |first3=Hal S.
 |last4=Rubin |first4=Donald B.
 |title=Bayesian Data Analysis
 |year=2003
 |edition=2nd
 |publisher=Chapman & Hall/CRC
 |location=Boca Raton, FL, USA
 |isbn=1-58488-388-X
 |doi=
}}</ref>
<ref name=Clevenson1975>
{{citation
 |last1=Clevenson |first1=M. Lawrence
 |last2=Zidek |first2=James V.
 |title=Simultaneous Estimation of the Means of Independent Poisson Laws
 |journal=Journal of the American Statistical Association
 |year=1975 |volume=70 |issue=351 |pages=698–705
 |doi=10.1080/01621459.1975.10482497
 |jstor=2285958
}}</ref>
<ref name=Berger1985>
{{citation
 |last=Berger |first=James O.
 |title=Statistical Decision Theory and Bayesian Analysis
 |year=1985
 |edition=2nd
 |publisher=Springer-Verlag
 |location=New York, NJ, USA
 |isbn=978-0-387-96098-2    
 |doi=10.1007/978-1-4757-4286-2
}}</ref>
<ref name=Loukas1986>
{{citation 
 |last1=Loukas |first=Sotirios
 |last2=Kemp |first2=C. David
 |title=The Index of Dispersion Test for the Bivariate Poisson Distribution 
 |journal=Biometrics
 |year=1986 |volume=42 |issue=4 |pages=941–948
 |doi= 10.2307/2530708
 |jstor=2530708
}}</ref>
}}

=== Sources ===
{{refbegin}}
* {{citation
 |last1 = Ahrens |first1 = Joachim H.
 |last2 = Dieter |first2 = Ulrich
 |title = Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions
 |journal = Computing
 |year = 1974
 |volume = 12
 |issue = 3
 |pages = 223–246
 |doi = 10.1007/BF02293108
}}
* {{citation
 |last1 = Ahrens |first1 = Joachim H.
 |last2 = Dieter |first2 = Ulrich
 |title = Computer Generation of Poisson Deviates
 |journal = ACM Transactions on Mathematical Software
 |year = 1982
 |volume = 8
 |issue = 2
 |pages = 163–179
 |doi = 10.1145/355993.355997
}}
* {{citation
 |last1 = Evans |first1 = Ronald J.
 |last2 = Boersma |first2 = J.
 |last3 = Blachman |first3 = N. M.
 |last4 = Jagers |first4 = A. A.
 |title = The Entropy of a Poisson Distribution: Problem 87-6
 |journal = SIAM Review
 |year = 1988
 |volume = 30
 |issue = 2
 |pages = 314–317
 |doi = 10.1137/1030059
}}
{{refend}}

== Further reading ==
* {{cite journal | last = Shanmugam | first = Ramalingam | title = Informatics about fear to report rapes using bumped-up Poisson model | journal = American Journal of Biostatistics | volume = 3 | issue = 1 | pages = 17–29 | doi = 10.3844/amjbsp.2013.17.29 | date = 2013 | ref = harv }}

{{-}}
{{ProbDistributions|discrete-infinite}}

{{Authority control}}

[[Category:Articles with example pseudocode]]
[[Category:Conjugate prior distributions]]
[[Category:Factorial and binomial topics]]
[[Category:Poisson distribution]]
[[Category:Infinitely divisible probability distributions]]