In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. The moment generating function (MGF) of a random variable X is a function MX(s ) defined as MX(s)=E[esX]. Moment generating function: definition, existence, moments, other characteristics. Each probability distribution has a unique MGF, which means they are especially useful for solving problems like finding the distribution for. If you have Googled “Moment Generating Function” and the first, the second, and. Normal Distribution | Moment Generating Function | Mean & Variance By MGF.