Moment-Generating Functions

Introduces the moment-generating function (MGF) of a random variable, shows how to compute it via the expectation of $e^{tX}$ , and uses derivatives of the MGF at $t=0$ to recover moments such as $E[X]$ , $E[X^2]$ , and $\text{Var}(X)$ .

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Tutorial

Defining the Moment-Generating Function

The moment-generating function (MGF) of a random variable $X$ is the function

M_X(t) = E\!\left[e^{tX}\right],

defined for all values of $t$ in some open interval around $0$ where this expectation is finite.

For a continuous random variable $X$ with density $f(x)$ , this becomes

M_X(t) = \int_{-\infty}^{\infty} e^{tx} f(x)\, dx.

As a small example, let $X$ be uniformly distributed on $[0,1]$ , so $f(x) = 1$ for $0 \le x \le 1$ . Then for $t \neq 0$ ,

\begin{align*} M_X(t) &= \int_0^1 e^{tx}\, dx \\[3pt] &= \left[\frac{e^{tx}}{t}\right]_0^1 \\[3pt] &= \frac{e^t - 1}{t}. \end{align*}

Notice also that $M_X(0) = E[e^{0}] = 1$ for any random variable $X$ .