Inverse transformation method using cumulative distribution function

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Cumulative distribution function ▷ Inverse transformation method

Theorem: Let $U$ be a continuous random variable having a standard uniform distribution. Then, the random variable

\[\label{eq:cdf-itm} X = F_X^{-1}(U)\]

has a probability distribution characterized by the invertible cumulative distribution function $F_X(x)$.

Proof: With the cumulative distribution function of the continuous uniform distribution, the cumulative distribution function of the standard uniform distribution is

\[\label{eq:suni-cdf} U \sim \mathcal{U}(0,1) \quad \Rightarrow \quad F_U(u) = \mathrm{Pr}(U \leq u) = u \; .\]

Thus, the cumulative distribution function of the transformation $X = F_X^{-1}(U)$ can be derived as

\[\label{eq:cdf-itm-qed} \begin{split} &\hphantom{=} \;\; \mathrm{Pr}(X \leq x) \\ &= \mathrm{Pr}(F_X^{-1}(U) \leq x) \\ &= \mathrm{Pr}(U \leq F_X(x)) \\ &= F_U(F_X(x)) \\ &= F_X(x) \; . \end{split}\]

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Sources:

Wikipedia (2021): "Inverse transform sampling"; in: Wikipedia, the free encyclopedia, retrieved on 2021-04-07; URL: https://en.wikipedia.org/wiki/Inverse_transform_sampling#Proof_of_correctness.

Metadata: ID: P221 | shortcut: cdf-itm | author: JoramSoch | date: 2021-04-07, 08:47.