Inverse transform sampling反变换采样法

感谢上海交大黄晨博士的博客：https://blog.csdn.net/doublehhcc/article/details/81166502
亦感谢上海交大博士生许志钦（现为纽约大学克朗研究院博士后）为直观解释所作的贡献，令博主与黄晨同学恍然大悟。

Goal:

Let ${\displaystyle X}$ be a random variable whose distribution can be described by the cumulative distribution function ${\displaystyle F_{X}}$.
We want to generate values of ${\displaystyle X}$ which are distributed according to this distribution.

Algorithm:

The inverse transform sampling method works as follows:

• Generate a random number ${\displaystyle u}$ from the standard uniform distribution in the interval ${\displaystyle [0,1]}$ e.g. from ${\displaystyle U\sim Unif[0,1].}$
• Find the inverse of the desired CDF, e.g. ${\displaystyle F_{X}^{-1}(x)}$.
• Compute ${\displaystyle X=F_{X}^{-1}(u)}$. This random variablel ${\displaystyle X}$ computed has distribution ${\displaystyle F_{X}}$.

Expressed differently, given a continuous uniform variable ${\displaystyle U}$ in ${\displaystyle [0,1]}$and an invertible cumulative distribution function ${\displaystyle F_{X}}$, the random variable ${\displaystyle X=F_{X}^{-1}(U)}$ has distribution ${\displaystyle F_{X}}$ (or, ${\displaystyle X}$ is distributed ${\displaystyle F_{X}}$).

逆变换采样的直观解释：

参考链接：https://en.wikipedia.org/wiki/Inverse_transform_sampling