# Inverse transform sampling

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Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the 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 $u$ between 0 and 1, interpreted as a probability, and then returns the smallest number $x\in \mathbb {R}$ such that $F(x)\geq u$ for the cumulative distribution function $F$ of a random variable. For example, imagine that $F$ 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.

Table info: u {\d...
Transformation from uniform sample to normal
$u$ $F^{-1}(u)$ .50
.9751.95996
.9952.5758
.9999994.75342
1-2−528.12589
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