Chi-squared Distribution

This is the gamma distribution with \(L=0.0\) and \(S=2.0\) and \(\alpha=\nu/2\) where \(\nu\) is called the degrees of freedom. If \(Z_{1}\ldots Z_{\nu}\) are all standard normal distributions, then \(W=\sum_{k}Z_{k}^{2}\) has (standard) chi-square distribution with \(\nu\) degrees of freedom.

The standard form (most often used in standard form only) has support \(x\geq0\).

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

\[M\left(t\right)=\frac{\Gamma\left(\frac{\nu}{2}\right)}{\left(\frac{1}{2}-t\right)^{\nu/2}}\]

\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