# Definition:Moment Generating Function

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## Definition

Let $X$ be a random variable.

Then the **moment generating function** of $X$, $M_X$, is defined as:

- $\map {M_X} t = \expect {e^{t X} }$

for all $t$ such that this expectation exists.

## Also known as

The term **moment generating function** is commonly abbreviated to **MGF** or **M.G.F**.

## Also see

- Results about
**moment generating functions**can be found here.