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

for $0<b,0<c$ where $\mathrm{\Gamma }\left(c\right)$ is the Gamma function.

Some sources use other parametrizations for this distribution; they might describe this distribution as $\mathrm{Gamma}\left(c\&comma;b\right)$ or $\mathrm{Gamma}\left(c\&comma;\frac{1}{b}\right)$.

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

The gamma variate with scale parameter 1 and shape parameter c is equivalent to the Erlang variate with shape parameter c.

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

$\mathrm{with}\left(\mathrm{Statistics}\right)\&colon;$

$X≔\mathrm{RandomVariable}\left(\mathrm{GammaDistribution}\left(b\&comma;c\right)\right)\&colon;$

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

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

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

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

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

$bc$

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

$b^{2}c$

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.