The Poisson random variable is a discrete probability random variable with probability function given by:

subject to the following conditions:

The Quantile and CDF functions applied to a Poisson random variable use a sequence of iterations in order to converge upon 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.

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

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

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

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

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

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

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

$\mathrm{\lambda }$

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

$\mathrm{\lambda }$

$Y≔\mathrm{PoissonRandomVariable}\left(3\right)\&colon;$

$\mathrm{ProbabilityFunction}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{CumulativeDistributionFunction}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

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.

The Student[Statistics][PoissonRandomVariable] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.