moment generating function

Given a random variable $X$, the moment generating function of $X$ is the following function:

$M_X(t)=E[e^{tX}]$ for $t\in R$ (if the expectation converges).

It can be shown that if the moment generating function of $X$ is defined on an interval around the origin, then

$E[X^k]={M_X^{(k)}(t)|}_{t=0}$

In other words, the $k$th-derivative of the moment generating function evaluated at zero is the $k$th moment of $X$.

Title	moment generating function
Canonical name	MomentGeneratingFunction
Date of creation	2013-03-22 11:53:51
Last modified on	2013-03-22 11:53:51
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Definition
Classification	msc 60E05
Classification	msc 46L05
Classification	msc 82-00
Classification	msc 83-00
Classification	msc 81-00
Related topic	CharacteristicFunction2
Related topic	CumulantGeneratingFunction