Gamma Distribution

The standard form for the gamma distribution is \(\left(\alpha>0\right)\) valid for \(x\geq0\) .

\begin{eqnarray*} f\left(x;\alpha\right) & = & \frac{1}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-x}\\ F\left(x;\alpha\right) & = & \frac{\gamma\left(\alpha,x\right)}{\Gamma(\alpha)}\\ G\left(q;\alpha\right) & = & \gamma^{-1}\left(\alpha,q\Gamma(\alpha)\right)\end{eqnarray*}

where \(\gamma\) is the lower incomplete gamma function, \(\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt\).

\[M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]

\begin{eqnarray*} \mu & = & \alpha\\ \mu_{2} & = & \alpha\\ \gamma_{1} & = & \frac{2}{\sqrt{\alpha}}\\ \gamma_{2} & = & \frac{6}{\alpha}\\ m_{d} & = & \alpha-1\end{eqnarray*}

\[h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right)\]

where

\[\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]

Implementation: scipy.stats.gamma