Moment Generating Function of Linear Transformation of Random Variable
Jump to navigation
Jump to search
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp {\beta t} \expect {\map \exp {\paren {\alpha t} X} }\) | Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\beta t} \map {M_X} {\alpha t}\) | Definition of Moment Generating Function |
$\blacksquare$