截断正态分布#
正态分布限制在由两个参数 \(A\) 和 \(B\) 给定的范围内。注意，这些 \(A\) 和 \(B\) 对应于标准形式中 \(x\) 的边界。对于 \(x\in\left[A,B\right]\)，我们得到
\begin{eqnarray*} f\left(x;A,B\right) & = & \frac{\phi\left(x\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\ F\left(x;A,B\right) & = & \frac{\Phi\left(x\right)-\Phi\left(A\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\ G\left(q;A,B\right) & = & \Phi^{-1}\left(q\Phi\left(B\right)+\Phi\left(A\right)\left(1-q\right)\right)\end{eqnarray*}
其中
\begin{eqnarray*} \phi\left(x\right) & = & \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\\ \Phi\left(x\right) & = & \int_{-\infty}^{x}\phi\left(u\right)du.\end{eqnarray*}
\begin{eqnarray*} \mu & = & \frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\\ \mu_{2} & = & 1+\frac{A\phi\left(A\right)-B\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}-\left(\frac{\phi\left(A\right)-\phi\left(B\right)}{\Phi\left(B\right)-\Phi\left(A\right)}\right)^{2}\end{eqnarray*}