麦克斯韦分布#
这是卡方分布的特例，其中 \(L=0\) 且 \(S=\frac{1}{\sqrt{a}}\) 以及 \(\nu=3.\) 它的支持范围是 \(x\geq0\).
\begin{eqnarray*} f\left(x\right) & = & \sqrt{\frac{2}{\pi}}x^{2}e^{-x^{2}/2}\\ F\left(x\right) & = & \frac{\gamma\left(\frac{3}{2},\frac{x^2}{2}\right)}{\Gamma(\frac{3}{2})}\\ G\left(q\right) & = & \sqrt{2\gamma^{-1}\left(\frac{3}{2},q\Gamma(\frac{3}{2})\right)}\end{eqnarray*}
\begin{eqnarray*} \mu & = & 2\sqrt{\frac{2}{\pi}}\\ \mu_{2} & = & 3-\frac{8}{\pi}\\ \gamma_{1} & = & \sqrt{2}\frac{32-10\pi}{\left(3\pi-8\right)^{3/2}}\\ \gamma_{2} & = & \frac{-12\pi^{2}+160\pi-384}{\left(3\pi-8\right)^{2}}\\ m_{d} & = & \sqrt{2}\\ m_{n} & = & \sqrt{2\gamma^{-1}\left(\frac{3}{2},\frac{1}{2}\Gamma(\frac{3}{2})\right)}\end{eqnarray*}
\[h\left[X\right]=\log\left(\sqrt{\frac{2\pi}{e}}\right)+\gamma.\]