= Poisson Distribution =

The '''Poisson distribution''' is a probability mass function.

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== Description ==

For a fixed number of independent trials, the number of successes is described by the [[Analysis/BinomialDistribution|binomial distribution]]. The probability of a success is denoted ''p'' in this case.

Instead of a fixed number of trials in some span of time, consider dividing the span of time into variable ''n'' intervals of size ''1/n''. The expected number of successes is denoted ''λ'', so the probability of success in one interval is ''λ/n''. As an example, if ''λ'' is the expected number of successes per day and ''n = 24'', then ''λ/n'' is the probability of success in one hour.

It follows that, for a given ''n'', the probability of ''0 ≤ k ≤ n'' successes is given by a binomial distribution.

{{attachment:dist1.svg}}

Take the limit of this distribution as ''n'' goes to infinity to arrive at the Poisson distribution. (This inherently means that ''p'' approaches zero as well.)

{{attachment:dist2.svg}}

The distribution describes the probability of ''k'' event successes.

{{attachment:dist3.svg}}

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = λ''.

The [[Analysis/MomentGeneratingFunctions|moment generating function]] of the Poisson distribution is:

{{attachment:mgf1.svg}}

This is now a standard power series that will always converge.

{{attachment:mgf2.svg}}

It follows that the expected value is ''λ''.

{{attachment:mgf3.svg}}

{{attachment:mgf4.svg}}

[[Analysis/Variance|Variance]] is given as ''Var[X] = λ''.

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== Usage ==



=== Poisson Approximation ===

A [[Analysis/BinomialDistribution|binomial distribution]] with a small ''p'' can be approximated by the Poisson distribution with ''λ = np''. This is mostly only useful for extremely large ''n'' such that computation of the factorial (for the binomial coefficient) would be computationally prohibitive.



=== Sum of Poisson Variables ===

The sum of multiple Poisson distributed random variables is also a Poisson variable.

If ''Y = X,,1,, + ... + X,,k,,'' where ''X,,i,, ~ Poisson(λ,,i,,)'' for all ''i'', then ''Y ~ Poisson(λ,,1,, + ... + λ,,k,,)''.

This can be proven using the [[Analysis/MomentGeneratingFunctions|moment generating function]]. Consider the simplest case of the sum of two variables:

{{attachment:sum.svg}}



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