9.1  What is an MGF?
9.1  What is an MGF? Moment generating function of \(X\)

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 \(h<t<h\).
Example 91
What is the moment generating function of a binomial random variable \(X\)?
Once we find the moment generating function of a random variable, we can use it to... tada!... generate moments!