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.. _continuous-chi2:
Chi-squared Distribution
========================
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`
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