半正态分布#
这是 \(L=a\) 和 \(S=b\) 以及 \(\nu=1.\) 的卡方分布的特例。这也是折叠正态分布的特例，形状参数为 \(c=0\) 且 \(S=S.\) 如果 \(Z\) 是（标准）正态分布的，则 \(\left|Z\right|\) 是半正态分布的。标准形式为
\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*}