Inverse transform sampling

Creator

Creator

Created

Created

2024 Nov 27 16:33

Editor

Editor

Edited

Edited

2025 Apr 29 0:39

Refs

Refs

Inversion sampling, inverse probability integral transform

Need to know the CDF analytically and have a tractable inverse

F_X^{-1}

.

Compute
Cumulative Distribution Function $F_X$

Generate random samples from
Uniform distribution $U \sim \mathcal{U}(0, 1)$

Inverse transform using $x=F^{-1} (u)$ to sample $x$ then $V = F_X^{-1} (U) = X$

The distribution of the random variable

V

is the same as

X

.

Proof

First of all,

F_X(X) \sim \mathcal{U}(0,1)

\text{CDF}(V) = P(F_X^{-1}(U) < t) = P(U < F_X(t)) = \int_{-\infty}^{F_X(t)} f_U(z) \, dz = F_X(t).

Inverse transform sampling

Inverse transform sampling 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 transform sampling

https://en.wikipedia.org/wiki/Inverse_transform_sampling

Inverse transform sampling

Backlinks

Cumulative Distribution Function

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