Definition 5.5.1. Moment Generating Function.
Given a probability function \(f(x)\text{,}\) the moment generating function is a transformation given by
\begin{equation*}
M(t) = E[e^{tx}]
\end{equation*}
where the expected value is a summation or integral dependent upon the nature of the random variable x. If the expected value does not exist (due perhaps to a f(x) with asymptotes) then the M(t) does not exist.