# HalfNormal Distribution¶

This is a special case of the chi distribution with $$L=a$$ and $$S=b$$ and $$\nu=1.$$ This is also a special case of the folded normal with shape parameter $$c=0$$ and $$S=S.$$ If $$Z$$ is (standard) normally distributed then, $$\left|Z\right|$$ is half-normal. The standard form is

\begin{eqnarray*} f\left(x\right) & = & \sqrt{\frac{2}{\pi}}e^{-x^{2}/2}I_{\left(0,\infty\right)}\left(x\right)\\ F\left(x\right) & = & 2\Phi\left(x\right)-1\\ G\left(q\right) & = & \Phi^{-1}\left(\frac{1+q}{2}\right)\end{eqnarray*}
$M\left(t\right)=\sqrt{2\pi}e^{t^{2}/2}\Phi\left(t\right)$
\begin{eqnarray*} \mu & = & \sqrt{\frac{2}{\pi}}\\ \mu_{2} & = & 1-\frac{2}{\pi}\\ \gamma_{1} & = & \frac{\sqrt{2}\left(4-\pi\right)}{\left(\pi-2\right)^{3/2}}\\ \gamma_{2} & = & \frac{8\left(\pi-3\right)}{\left(\pi-2\right)^{2}}\\ m_{d} & = & 0\\ m_{n} & = & \Phi^{-1}\left(\frac{3}{4}\right)\end{eqnarray*}
\begin{eqnarray*} h\left[X\right] & = & \log\left(\sqrt{\frac{\pi e}{2}}\right)\\ & \approx & 0.72579135264472743239.\end{eqnarray*}

Implementation: scipy.stats.halfnorm

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