# 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