HalfNormal DistributionΒΆ
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}\\
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