Moment-generating function of a function of a random variable

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Moment-generating function of arbitrary function

Theorem: Let $X$ be a random variable with the expected value function $\mathrm{E}_X[\cdot]$. Then, the moment-generating function of $Y = g(X)$ is equal to

\[\label{eq:mgf-fct} M_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(t \, g(X)) \right] \; .\]

Proof: The moment-generating function is defined as

\[\label{eq:mgf} M_Y(t) = \mathrm{E} \left[ \mathrm{exp}(t \, Y) \right] \; .\]

Due of the law of the unconscious statistician

\[\label{eq:mean-lotus} \begin{split} \mathrm{E}[g(X)] &= \sum_{x \in \mathcal{X}} g(x) f_X(x) \\ \mathrm{E}[g(X)] &= \int_{\mathcal{X}} g(x) f_X(x) \, \mathrm{d}x \; , \end{split}\]

$Y = g(X)$ can simply be substituted into \eqref{eq:mgf} to give

\[\label{eq:mgf-fct-qed} M_Y(t) = \mathrm{E}_X \left[ \mathrm{exp}(t \, g(X)) \right] \; .\]

∎

Sources:

Taboga, Marco (2017): "Functions of random vectors and their distribution"; in: Lectures on probability and mathematical statistics, retrieved on 2021-09-22; URL: https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors.

Metadata: ID: P260 | shortcut: mgf-fct | author: JoramSoch | date: 2021-09-22, 09:00.