This is documentation for an old release of SciPy (version 1.5.0). Read this page in the documentation of the latest stable release (version 1.15.1).
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