Overview Section
The expected values \(E(X), E(X^2), E(X^3), \ldots, \text{and } E(X^r)\) are called moments. As you have already experienced in some cases, the mean:
\(\mu=E(X)\)
and the variance:
\(\sigma^2=\text{Var}(X)=E(X^2)-\mu^2\)
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.
In this lesson, we'll first learn what a moment-generating function is, and then we'll earn how to use moment generating functions (abbreviated "m.g.f."):
- to find moments and functions of moments, such as \(\mu\) and \(\sigma^2\)
- to identify which probability mass function a random variable \(X\) follows
Objectives
Upon completion of this lesson, you should be able to:
- To learn the definition of a moment-generating function.
- To find the moment-generating function of a binomial random variable.
- To learn how to use a moment-generating function to find the mean and variance of a random variable.
- To learn how to use a moment-generating function to identify which probability mass function a random variable \(X\) follows.
- To understand the steps involved in each of the proofs in the lesson.
- To be able to apply the methods learned in the lesson to new problems.