The gamma random variable is a continuous probability random variable with probability density function given by:

subject to the following conditions:

The gamma variate with scale parameter b and shape parameter 1 is equivalent to the Exponential variate with scale parameter b.

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

$X≔\mathrm{GammaRandomVariable}\left(b\,c\right)\:$

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

$\left\{\begin{array}{cc}0& u<0\\ \frac{{\left(\frac{u}{b}\right)}^{c-1}{\&ExponentialE;}^{-\frac{u}{b}}}{b\mathrm{\Gamma }\left(c\right)}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{{\left(\frac{0.5}{b}\right)}^{c-1.}{\&ExponentialE;}^{-\frac{0.5}{b}}}{b\mathrm{\Gamma }\left(c\right)}$

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

$bc$

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

$b^{2}c$

$Y≔\mathrm{GammaRandomVariable}\left(2\&comma;3\right)\&colon;$

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

$\mathrm{CDF}\left(Y\&comma;x\right)$

$\left\{\begin{array}{cc}0& x\le 0\\ -\frac{x^2{\&ExponentialE;}^{-\frac{x}{2}}}{8}-\frac{{\&ExponentialE;}^{-\frac{x}{2}}x}{2}-{\&ExponentialE;}^{-\frac{x}{2}}+1& 0<x\end{array}\right.$

$\mathrm{CDF}\left(Y\&comma;6\&comma;\mathrm{output}=\mathrm{plot}\right)$

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][GammaRandomVariable] command was introduced in Maple 18.

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