学生 t 分布#
有一个形状参数 \(\nu>0\)，其支持范围为 \(x\in\mathbb{R}\).
\begin{eqnarray*} f\left(x;\nu\right) & = & \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi\nu}\Gamma\left(\frac{\nu}{2}\right)\left[1+\frac{x^{2}}{\nu}\right]^{\frac{\nu+1}{2}}}\\ F\left(x;\nu\right) & = & \left\{ \begin{array}{ccc} \frac{1}{2}I\left(\frac{\nu}{\nu+x^{2}}; \frac{\nu}{2},\frac{1}{2}\right) & & x\leq0\\ 1-\frac{1}{2}I\left(\frac{\nu}{\nu+x^{2}}; \frac{\nu}{2},\frac{1}{2}\right) & & x\geq0 \end{array} \right.\\ G\left(q;\nu\right) & = & \left\{ \begin{array}{ccc} -\sqrt{\frac{\nu}{I^{-1}\left(2q; \frac{\nu}{2},\frac{1}{2}\right)}-\nu} & & q\leq\frac{1}{2}\\ \sqrt{\frac{\nu}{I^{-1}\left(2-2q; \frac{\nu}{2},\frac{1}{2}\right)}-\nu} & & q\geq\frac{1}{2} \end{array} \right. \end{eqnarray*}
\begin{eqnarray*} m_{n}=m_{d}=\mu & = & 0\\ \mu_{2} & = & \frac{\nu}{\nu-2}\quad\nu>2\\ \gamma_{1} & = & 0\quad\nu>3\\ \gamma_{2} & = & \frac{6}{\nu-4}\quad\nu>4\end{eqnarray*}
其中 \(I\left(x; a,b\right)\) 是不完全贝塔积分，且 \(I^{-1}\left(I\left(x; a,b\right); a,b\right)=x\)。当 \(\nu\rightarrow\infty,\) 此分布逼近标准正态分布。
\[h\left[X\right]=\frac{\nu+1}{2} \left[\psi \left(\frac{1+\nu}{2} \right) -\psi \left(\frac{\nu}{2} \right) \right] + \ln \left[ \sqrt{\nu} B \left( \frac{\nu}{2}, \frac{1}{2} \right) \right]\]
其中 \(\psi(x)\) 是双伽马函数，\(B(x, y)\) 是贝塔函数。
参考文献#
“学生 t 分布”，维基百科，https://en.wikipedia.org/wiki/Student%27s_t-distribution