Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability functions ▷ Distributional transformation

Theorem: Let $X$ and $Y$ be two continuous random variables with cumulative distribution function $F_X(x)$ and invertible cumulative distribution function $F_Y(y)$. Then, the random variable

\[\label{eq:cdf-dt} \tilde{X} = F_Y^{-1}(F_X(X))\]

has the same probability distribution as $Y$.

Proof: The cumulative distribution function of the transformation $\tilde{X} = F_Y^{-1}(F_X(X))$ can be derived as

\[\label{eq:cdf-dt-qed} \begin{split} F_{\tilde{X}}(y) &= \mathrm{Pr}\left( \tilde{X} \leq y \right) \\ &= \mathrm{Pr}\left( F_Y^{-1}(F_X(X)) \leq y \right) \\ &= \mathrm{Pr}\left( F_X(X) \leq F_Y(y) \right) \\ &= \mathrm{Pr}\left( X \leq F_X^{-1}(F_Y(y)) \right) \\ &= F_X\left( F_X^{-1}(F_Y(y)) \right) \\ &= F_Y(y) \\ \end{split}\]

which shows that $\tilde{X}$ and $Y$ have the same cumulative distribution function and are thus identically distributed.

Sources:

Metadata: ID: P222 | shortcut: cdf-dt | author: JoramSoch | date: 2021-04-07, 09:19.