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Rayleigh Distribution¶
This is Chi distribution with \(L=0.0\) and \(\nu=2\) and \(S=S\) (no location parameter is generally used), the mode of the distribution is \(S.\)
\begin{eqnarray*} f\left(r\right) & = & re^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\ F\left(r\right) & = & 1-e^{-r^{2}/2}I_{[0,\infty)}\left(x\right)\\ G\left(q\right) & = & \sqrt{-2\log\left(1-q\right)}\end{eqnarray*}
\begin{eqnarray*} \mu & = & \sqrt{\frac{\pi}{2}}\\ \mu_{2} & = & \frac{4-\pi}{2}\\ \gamma_{1} & = & \frac{2\left(\pi-3\right)\sqrt{\pi}}{\left(4-\pi\right)^{3/2}}\\ \gamma_{2} & = & \frac{24\pi-6\pi^{2}-16}{\left(4-\pi\right)^{2}}\\ m_{d} & = & 1\\ m_{n} & = & \sqrt{2\log\left(2\right)}\end{eqnarray*}
\[h\left[X\right]=\frac{\gamma}{2}+\log\left(\frac{e}{\sqrt{2}}\right).\]
\[\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(\frac{n}{2}+1\right)\]
Implementation: scipy.stats.rayleigh