截断正态分布¶
正态分布被限制在由两个参数给出的一定范围内的正态分布 \(A\) 和 \(B\) 。请注意，这是 \(A\) 和 \(B\) 对应于 \(x\) 在标准形式下。为 \(x\in\left[A,B\right]\) 我们会得到
\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*}
哪里
\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{等式*}\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*}