# Definition:Moment Generating Function

## 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.