The Poisson distribution is a discrete probability distribution with probability function given by:

subject to the following conditions:

Note that the Poisson command is inert and should be used in combination with the RandomVariable command.

The Quantile and CDF functions applied to a Poisson distribution use a sequence of iterations in order to converge on the desired output point.  The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Poisson}\left(\mathrm{\lambda }\right)\right)\:$

$\mathrm{ProbabilityFunction}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{{\mathrm{\lambda }}^u{\&ExponentialE;}^{-\mathrm{\lambda }}}{u!}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ProbabilityFunction}\left(X\&comma;2\right)$

$\frac{{\mathrm{\lambda }}^2{\&ExponentialE;}^{-\mathrm{\lambda }}}{2}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{\lambda }$

$\mathrm{Variance}\left(X\right)$

$\mathrm{\lambda }$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.