Inverse transform sampling

The key to inversion sampling is the probability integral transform, which says that for a random variable \(X\) with CDF \(F_X\), \(F_X(X)\) is uniformly distributed. Or \(U \sim F_X(X)\).

If \(F_X\) is invertible, we have \(F_X^{-1}(U) \sim X\). This allows us to draw samples from \(X\) if we can draw samples from \(U\).