Moment Generating Function of Linear Combination of Independent Random Variables

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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:

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

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

Then:

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

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


Proof

\(\ds \map {M_X} t\) \(=\) \(\ds \expect {\map \exp {t X} }\) Definition of Moment Generating Function
\(\ds \) \(=\) \(\ds \expect {\map \exp {t \sum_{i \mathop = 1}^n k_i X_i} }\)
\(\ds \) \(=\) \(\ds \expect {\prod_{i \mathop = 1}^n \map \exp {t k_i X_i} }\) Exponential of Sum
\(\ds \) \(=\) \(\ds \prod_{i \mathop = 1}^n \expect {\map \exp {t k_i X_i} }\) Condition for Independence from Product of Expectations
\(\ds \) \(=\) \(\ds \prod_{i \mathop = 1}^n \map {M_{X_i} } {k_i t}\) Definition of Moment Generating Function

$\blacksquare$