# 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 9-1

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... ta-da!... generate moments!

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