Constructing Moment-Generating Functions for Discrete Probability Distributions

Construct moment-generating functions (MGFs) for common discrete distributions -- Bernoulli, Binomial, Poisson, and Discrete Uniform -- by applying the definition $M_X(t)=E[e^{tX}]$ and using algebraic identities (the binomial theorem and the Taylor series of the exponential).

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Tutorial

The MGF of a Discrete Random Variable

The moment-generating function (MGF) of a discrete random variable $X$ with probability mass function $p_X$ is defined as

M_X(t) = E[e^{tX}] = \sum\limits_{x} e^{tx}\, p_X(x),

where the sum runs over every $x$ in the support of $X.$

Notice that $M_X(t)$ is a function of the real variable $t$ alone -- the random variable $X$ has been summed out.

For example, suppose $X$ takes values $0,$ $1,$ and $2$ with probabilities $\dfrac{1}{2},$ $\dfrac{1}{3},$ and $\dfrac{1}{6},$ respectively. Then

\begin{align*} M_X(t) &= e^{0\cdot t}\cdot\dfrac{1}{2} + e^{1\cdot t}\cdot\dfrac{1}{3} + e^{2t}\cdot\dfrac{1}{6} \\[3pt] &= \dfrac{1}{2} + \dfrac{e^{t}}{3} + \dfrac{e^{2t}}{6}. \end{align*}

To construct the MGF of a named distribution, we plug its pmf into this sum and simplify.