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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