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

Theorem: Let $X_1, \ldots, X_n$ be $n$ independent random variables with moment-generating functions $M_{X_i}(t)$. Then, the moment-generating function of the linear combination $X = \sum_{i=1}^{n} a_i X_i$ is given by

\[\label{eq:mgf-lincomb} M_X(t) = \prod_{i=1}^{n} M_{X_i}(a_i t)\]

where $a_1, \ldots, a_n$ are $n$ real numbers.

Proof: The moment-generating function of a random variable $X_i$ is

\[\label{eq:mfg} M_{X_i}(t) = \mathrm{E} \left( \exp \left[ t X_i \right] \right)\]

and therefore the moment-generating function of the linear combination $X$ is given by

\[\label{eq:mgf-lincomb-s1} \begin{split} M_X(t) &= \mathrm{E} \left( \exp \left[ t X \right] \right) \\ &= \mathrm{E} \left( \exp \left[ t \sum_{i=1}^{n} a_i X_i \right] \right) \\ &= \mathrm{E} \left( \prod_{i=1}^{n} \exp \left[ t \, a_i X_i \right] \right) \; . \end{split}\]

Because the expected value is multiplicative for independent random variables, we have

\[\label{eq:mgf-lincomb-s2} \begin{split} M_X(t) &= \prod_{i=1}^{n} \mathrm{E} \left( \exp \left[ (a_i t) X_i \right] \right) \\ &= \prod_{i=1}^{n} M_{X_i}(a_i t) \; . \end{split}\]
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Metadata: ID: P155 | shortcut: mgf-lincomb | author: JoramSoch | date: 2020-08-19, 08:36.