Moment-generating function of a sum of independent random variables

Statement

Let $X_1, \dots, X_n$ be independent real random variables, and let $S = X_1 + \dots + X_n$. Then for each $t \in \mathbb{R}$, the moment-generating functions satisfy $$M_S(t) = M_{X_1}(t) \, M_{X_2}(t) \cdots M_{X_n}(t),$$ in $(0, \infty]$.

Remarks

Proof: $e^{tS} = e^{tX_1} \cdots e^{tX_n}$. The factors are nonnegative and independent, so the product rule for expectations applies. For i.i.d. $X_i$, $M_S(t) = M_{X_1}(t)^n$. Example: if $X_i \sim \operatorname{Exp}(\lambda)$ are independent, then $M_S(t) = (\lambda/(\lambda - t))^n$ for $t < \lambda$. This is the moment-generating function of the gamma distribution $\operatorname{Gamma}(n, \lambda)$.

Explanations

L1 human

When you add independent random variables, their moment-generating functions multiply. This makes it easy to identify the law of a sum, if you know which law has the product as its moment-generating function.