This is documentation for an old release of SciPy (version 1.2.1). Read this page in the documentation of the latest stable release (version 1.15.1).
Chi Distribution¶
Generated by taking the (positive) square-root of chi-squared variates.
\begin{eqnarray*} f\left(x;\nu\right) & = & \frac{x^{\nu-1}e^{-x^{2}/2}}{2^{\nu/2-1}\Gamma\left(\frac{\nu}{2}\right)}I_{\left(0,\infty\right)}\left(x\right)\\ F\left(x;\nu\right) & = & \Gamma\left(\frac{\nu}{2},\frac{x^{2}}{2}\right)\\ G\left(\alpha;\nu\right) & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\alpha\right)}\end{eqnarray*}
\[M\left(t\right)=\Gamma\left(\frac{v}{2}\right)\,_{1}F_{1}\left(\frac{v}{2};\frac{1}{2};\frac{t^{2}}{2}\right)+\frac{t}{\sqrt{2}}\Gamma\left(\frac{1+\nu}{2}\right)\,_{1}F_{1}\left(\frac{1+\nu}{2};\frac{3}{2};\frac{t^{2}}{2}\right)\]
\begin{eqnarray*} \mu & = & \frac{\sqrt{2}\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}\\ \mu_{2} & = & \nu-\mu^{2}\\ \gamma_{1} & = & \frac{2\mu^{3}+\mu\left(1-2\nu\right)}{\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{2\nu\left(1-\nu\right)-6\mu^{4}+4\mu^{2}\left(2\nu-1\right)}{\mu_{2}^{2}}\\ m_{d} & = & \sqrt{\nu-1}\quad\nu\geq1\\ m_{n} & = & \sqrt{2\Gamma^{-1}\left(\frac{\nu}{2},\frac{1}{2}\right)}\end{eqnarray*}
Implementation: scipy.stats.chi