9.1 - What is an MGF?

Let \(X\) be a discrete random variable with probability mass function \(f(x)\) and support \(S\). Then:

\(M(t)=E(e^{tX})=\sum\limits_{x\in S} e^{tx}f(x)\)

is the moment generating function of \(X\) as long as the summation is finite for some interval of \(t\) around 0. That is, \(M(t)\) is the moment generating function ("m.g.f.") of \(X\) if there is a positive number \(h\) such that the above summation exists and is finite for \(-h<t<h\).