Why do we use moment generating functions?

What is the moment-generating function of a random variable?

A moment-generating function uniquely determines the probability distribution of a random variable. If the support S is { b 1, b 2, b 3, … }, then the moment-generating function: Therefore, the coefficient of:

Besides helping to find moments, the moment generating function has an important property often called the uniqueness property. The uniqueness property means that, if the mgf exists for a random variable, then there one and only one distribution associated with that mgf.

The expected values E ( X), E ( X 2), E ( X 3), …, and E ( X r) are called moments. As you have already experienced in some cases, the mean: which are functions of moments, are sometimes difficult to find. Special functions, called moment-generating functions can sometimes make finding the mean and variance of a random variable simpler.