Constructing Moment-Generating Functions for Continuous Probability Distributions

Construct moment-generating functions (MGFs) for the continuous uniform, exponential, normal, and gamma (including chi-square) distributions by evaluating $E[e^{tX}]$ as an integral against the density.

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Tutorial

Introduction

For a continuous random variable $X$ with probability density function $f(x),$ the moment-generating function (MGF) is

M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f(x)\,dx,

defined for all real $t$ for which the integral converges.

This is the direct analog of the discrete MGF: the sum $\sum_x e^{tx} p(x)$ is replaced by an integral against the density. As in the discrete case, the function is moment-generating because

E[X^n] = M_X^{(n)}(0).

To construct an MGF for a specific distribution, we substitute the density into the integral, simplify, and record the values of $t$ that keep the integral finite.