高斯超几何分布#
四个形状参数为 \(\alpha>0\)，\(\beta>0\)，\(-\infty < \gamma < \infty\)，以及 \(z > -1\)。支持范围为 \(x\in\left[0,1\right]\).
\[\text{Let }C=\frac{1}{B\left(\alpha,\beta\right)\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)}\]
\begin{eqnarray*} f\left(x;\alpha,\beta,\gamma,z\right) & = & Cx^{\alpha-1}\frac{\left(1-x\right)^{\beta-1}}{\left(1+zx\right)^{\gamma}}\\ \mu_{n}^{\prime} & = & \frac{B\left(n+\alpha,\beta\right)}{B\left(\alpha,\beta\right)}\frac{\,_{2}F_{1}\left(\gamma,\alpha+n;\alpha+\beta+n;-z\right)}{\,_{2}F_{1}\left(\gamma,\alpha;\alpha+\beta;-z\right)}\end{eqnarray*}