Poisson

The Poisson cumulative distribution function is a discrete function defined for non-negative integers x. The Poisson distribution with mean μ > 0 is defined by the summation

$\displaystyle \func$PoissonDist$\displaystyle \left(\vphantom{ x;\mu }\right.$x;μ$\displaystyle \left.\vphantom{ x;\mu }\right)$ = $\displaystyle \sum_{{k=0}}^{{x}}$$\displaystyle {\frac{{\mu ^{k}e^{-\mu }%
}}{{k!}}}$

The Poisson probability density function is

$\displaystyle \limfunc$PoissonDen$\displaystyle \left(\vphantom{ k;\mu }\right.$k;μ$\displaystyle \left.\vphantom{ k;\mu }\right)$ = $\displaystyle {\frac{{\mu ^{k}e^{-\mu }}}{{k!}}}$

for nonnegative integers k and real numbers μ > 0. A Poisson distribution table lists selected values of the Poisson probability density function $\limfunc$PoissonDen$\left(\vphantom{ k;\mu }\right.$k;μ$\left.\vphantom{ k;\mu }\right)$.

$\blacktriangleright$ Evaluate Numerically

$\limfunc$PoissonDen$\left(\vphantom{ 2;3}\right.$2;3$\left.\vphantom{ 2;3}\right)$ = .22404

$\limfunc$PoissonDen$\left(\vphantom{ 5;0.3}\right.$5;0.3$\left.\vphantom{ 5;0.3}\right)$ = 1.5002×10-5

$\limfunc$PoissonDen$\left(\vphantom{ 10;4}\right.$10;4$\left.\vphantom{ 10;4}\right)$ = 5.2925×10-3

The Poisson distribution can be used to approximate the binomial distribution when the probability is small and n is large; that is,

$\displaystyle \func$PoissonDist(k;μ) $\displaystyle \approx$ $\displaystyle \func$BinomialDist$\displaystyle \left(\vphantom{ k;\mu ,\mu
\left( 1-p\right) }\right.$k;μ, μ$\displaystyle \left(\vphantom{ 1-p}\right.$1 - p$\displaystyle \left.\vphantom{ 1-p}\right)$$\displaystyle \left.\vphantom{ k;\mu ,\mu
\left( 1-p\right) }\right)$

where μ = np. This distribution has been used as a model for a variety of random phenomena of practical importance.