doc/source/tutorial/stats/continuous_chi2.rst
.. _continuous-chi2:
This is the gamma distribution with :math:L=0.0 and :math:S=2.0 and :math:\alpha=\nu/2 where :math:\nu is called the degrees of freedom. If :math:Z_{1}, \ldots, Z_{\nu} are all standard normal distributions, then :math:W=\sum_{k}Z_{k}^{2} has (standard) chi-square distribution with :math:\nu degrees of freedom.
The standard form (most often used in standard form only) has support :math:x\geq0.
.. math:: :nowrap:
\begin{eqnarray*} f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\frac{\nu}{2}\right)}\left(\frac{x}{2}\right)^{\nu/2-1}e^{-x/2}\\
F\left(x;\alpha\right) & = & \frac{\gamma\left(\frac{\nu}{2},\frac{x}{2}\right)}{\Gamma(\frac{\nu}{2})}\\
G\left(q;\alpha\right) & = & 2\gamma^{-1}\left(\frac{\nu}{2},q{\Gamma(\frac{\nu}{2})}\right)\end{eqnarray*}
where :math:\gamma is the lower incomplete gamma function, :math:\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt.
.. math::
M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}}
.. math:: :nowrap:
\begin{eqnarray*} \mu & = & \nu\\
\mu_{2} & = & 2\nu\\
\gamma_{1} & = & \frac{2\sqrt{2}}{\sqrt{\nu}}\\
\gamma_{2} & = & \frac{12}{\nu}\\
m_{d} & = & \frac{\nu}{2}-1\end{eqnarray*}
Implementation: scipy.stats.chi2