Probability integral transform using cumulative distribution function

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Cumulative distribution function ▷ Probability integral transform

Theorem: Let $X$ be a continuous random variable with invertible cumulative distribution function $F_X(x)$. Then, the random variable

\[\label{eq:cdf-pit} Y = F_X(X)\]

has a standard uniform distribution.

Proof: The cumulative distribution function of $Y = F_X(X)$ can be derived as

\[\label{eq:cdf-pit-qed} \begin{split} F_Y(y) &= \mathrm{Pr}(Y \leq y) \\ &= \mathrm{Pr}(F_X(X) \leq y) \\ &= \mathrm{Pr}(X \leq F_X^{-1}(y)) \\ &= F_X(F_X^{-1}(y)) \\ &= y \\ \end{split}\]

which is the cumulative distribution function of a continuous uniform distribution with $a = 0$ and $b = 1$, i.e. the cumulative distribution function of the standard uniform distribution $\mathcal{U}(0,1)$.

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Sources:

Wikipedia (2021): "Probability integral transform"; in: Wikipedia, the free encyclopedia, retrieved on 2021-04-07; URL: https://en.wikipedia.org/wiki/Probability_integral_transform#Proof.

Metadata: ID: P220 | shortcut: cdf-pit | author: JoramSoch | date: 2021-04-07, 08:47.