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Statistics[Distributions]

 Gamma
 gamma distribution

 Calling Sequence Gamma(b, c) GammaDistribution(b, c)

Parameters

 b - scale parameter c - shape parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{\left(\frac{t}{b}\right)}^{c-1}{ⅇ}^{-\frac{t}{b}}}{b\mathrm{\Gamma }\left(c\right)}& \mathrm{otherwise}\end{array}\right\$

 for $0 where $\mathrm{\Gamma }\left(c\right)$ is the Gamma function.
 • Some sources use other parametrizations for this distribution; they might describe this distribution as $\mathrm{Gamma}\left(c,b\right)$ or $\mathrm{Gamma}\left(c,\frac{1}{b}\right)$.
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{GammaDistribution}\left(b,c\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{\left(\frac{{u}}{{b}}\right)}^{{c}{-}{1}}{}{{ⅇ}}^{{-}\frac{{u}}{{b}}}}{{b}{}{\mathrm{\Gamma }}{}\left({c}\right)}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{\left(\frac{{0.5}}{{b}}\right)}^{{c}{-}{1.}}{}{{ⅇ}}^{{-}\frac{{0.5}}{{b}}}}{{b}{}{\mathrm{\Gamma }}{}\left({c}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${b}{}{c}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{b}}^{{2}}{}{c}$ (4)

References

 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.