Inverse transform sampling反变换采样法

感谢上海交大黄晨博士的博客：https://blog.****.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 X${\displaystyle X}$ be a random variable whose distribution can be described by the cumulative distribution function FX${\displaystyle F_X}$.
We want to generate values of X${\displaystyle X}$ which are distributed according to this distribution.

Algorithm:

The inverse transform sampling method works as follows:

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

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

逆变换采样的直观解释：

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