Inverse transform sampling

Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, golden rule) is a basic method for pseudo-random number sampling, i.e. for generating sample numbers at random from any probability distribution given its cumulative distribution function.

Inverse transformation sampling takes uniform samples of a number ${\displaystyle u}$ between 0 and 1, interpreted as a probability, and then returns the largest number ${\displaystyle x}$ from the domain of the distribution ${\displaystyle P(X)}$ such that ${\displaystyle P(-\infty <X<x)\leq u}$. For example, imagine that ${\displaystyle P(X)}$ is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution.

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