Moment-generating function

Definition

Let $X$ be a real random variable. The moment-generating function of $X$ is the function $$M_X(t) = E\left[e^{tX}\right], \quad t \in \mathbb{R},$$ with values in $(0, \infty]$.

Remarks

Always $M_X(0) = 1$. The set of $t$ with $M_X(t) < \infty$ is an interval that contains $0$. If $M_X$ is finite on an open interval around $0$, then all moments of $X$ are finite and $M_X$ determines the law of $X$. Some random variables have $M_X(t) = \infty$ for all $t \ne 0$. An example is a Cauchy random variable. For $X \sim \operatorname{Exp}(\lambda)$, $M_X(t) = \lambda / (\lambda - t)$ for $t < \lambda$, and $M_X(t) = \infty$ for $t \ge \lambda$.

Explanations

L1 human

The moment-generating function is a tool that turns a distribution into a function of one real variable. It is useful because sums of independent variables become products, and moments become derivatives.