Moment Generating Function of Linear Transformation of Random Variable

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Theorem

Let $X$ be a random variable.

Let $\alpha$ and $\beta$ be real numbers.

Let $Z = \alpha X + \beta$.

Let $M_X$ be the moment generating function of $X$.


Then the moment generating function of $Z$, $M_Z$, is given by:

$\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$


Proof

\(\ds \map {M_Z} t\) \(=\) \(\ds \expect {\map \exp {t Z} }\) Definition of Moment Generating Function
\(\ds \) \(=\) \(\ds \expect {\map \exp {t \paren {\alpha X + \beta} } }\)
\(\ds \) \(=\) \(\ds \expect {\map \exp {\paren {\alpha t} X} \map \exp {\beta t} }\) Exponential of Sum
\(\ds \) \(=\) \(\ds \map \exp {\beta t} \expect {\map \exp {\paren {\alpha t} X} }\) Linearity of Expectation Function
\(\ds \) \(=\) \(\ds e^{\beta t} \map {M_X} {\alpha t}\) Definition of Moment Generating Function

$\blacksquare$