Gamma

The Gamma distribution is defined for x > 0 by the integral

$\displaystyle \limfunc$GammaDist$\displaystyle \left(\vphantom{ x;a,b}\right.$x;a, b$\displaystyle \left.\vphantom{ x;a,b}\right)$ = $\displaystyle {\frac{{1}}{{b^{a}\Gamma \left( a\right)
}}}$$\displaystyle \int_{{0}}^{{x}}$ua-1e-$\scriptstyle {\frac{{u}}{{b}}}$du

where Γ$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = $\int_{{0}}^{{\infty }}$e-uut-1du is the Gamma function. The parameters a and b are called the shape parameter and scale parameter, respectively. The mean of this distribution is ab and the variance is ab2. The probability density function for the gamma distribution is

$\displaystyle \limfunc$GammaDen(u;a, b) = $\displaystyle {\frac{{1}}{{b^{a}\Gamma \left( a\right)
}}}$ua-1e-$\scriptstyle {\frac{{u}}{{b}}}$

The following plots show cumulative distribution functions $\limfunc$GammaDist$\left(\vphantom{ x;a,b}\right.$x;a, b$\left.\vphantom{ x;a,b}\right)$ and probability density functions $\limfunc$GammaDen(x;a, b) for (a, b) = $\left(\vphantom{ 1,.5}\right.$1,.5$\left.\vphantom{ 1,.5}\right)$,$\left(\vphantom{ 1,1}\right.$1, 1$\left.\vphantom{ 1,1}\right)$,$\left(\vphantom{
2,1}\right.$2, 1$\left.\vphantom{
2,1}\right)$ and 0≤x≤4.

itbpFU2.1577in1.7573in0inGamma density functions itbpFU2.1577in1.7573in0inGamma distribution functions