Moment Generating Function of Linear Combination of Independent Random Variables

Theorem

Let $X_1, X_2, \ldots, X_n$ be independent random variables.

Let $k_1, k_2, \ldots, k_n$ be real numbers.

Let:

$\ds X = \sum_{i \mathop = 1}^n k_i X_i$

Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \le i \le n$.

Then:

$\ds \map {M_X} t = \prod_{i \mathop = 1}^n \map {M_{X_i}} {k_i t}$

for all $t$ such that $M_{X_i}$ exists for all $1 \le i \le n$.

$\ds \map {M_X} t$

$=$

$\ds \expect {\map \exp {t X} }$

$\ds $

$=$

$\ds \expect {\map \exp {t \sum_{i \mathop = 1}^n k_i X_i} }$

$\ds $

$=$

$\ds \expect {\prod_{i \mathop = 1}^n \map \exp {t k_i X_i} }$

$\ds $

$=$

$\ds \prod_{i \mathop = 1}^n \expect {\map \exp {t k_i X_i} }$

$\ds $

$=$

$\ds \prod_{i \mathop = 1}^n \map {M_{X_i} } {k_i t}$

$\blacksquare$