Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability mass function ▷ Probability mass function of invertible function

Theorem: Let $X$ be an $n \times 1$ random vector of discrete random variables with possible outcomes $\mathcal{X}$ and let $g: \; \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an invertible function on the support of $X$. Then, the probability mass function of $Y = g(X)$ is given by

\[\label{eq:pmf-invfct} f_Y(y) = \left\{ \begin{array}{rl} f_X(g^{-1}(y)) \; , & \text{if} \; y \in \mathcal{Y} \\ 0 \; , & \text{if} \; y \notin \mathcal{Y} \end{array} \right.\]

where $g^{-1}(y)$ is the inverse function of $g(x)$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$:

\[\label{eq:Y-range} \mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace \; .\]

Proof: Because an invertible function is a one-to-one mapping, the probability mass function of $Y$ can be derived as follows:

\[\label{eq:pmf-invfct-qed} \begin{split} f_Y(y) &= \mathrm{Pr}(Y = y) \\ &= \mathrm{Pr}(g(X) = y) \\ &= \mathrm{Pr}(X = g^{-1}(y)) \\ &= f_X(g^{-1}(y)) \; . \end{split}\]
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Metadata: ID: P253 | shortcut: pmf-invfct | author: JoramSoch | date: 2021-08-30, 05:13.