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'moment generating function'
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| Title of object: |
moment generating function |
| Canonical Name: |
MomentGeneratingFunction |
| Type: |
Definition |
| Created on: |
2001-10-26 02:53:10 |
| Modified on: |
2004-03-08 14:15:11 |
| Classification: |
msc:60E05 |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{graphicx}
%\usepackage{xypic} |
Content:
Given a random variable $X$, the \emph{moment generating function} of $X$ is the following function:\\
\par
$M_X(t) = E[e^{tX}]$ for $t \in R$ (if the expectation converges).
\par
\par
It can be shown that if the moment generating function of X is defined on an interval around the origin, then\\
\par
$E[X^k] = M_X^{(k)}(t) |_{t=0} $\\
\par
In other words, the $k$th-derivative of the moment generating function evaluated at zero is the $k$th moment of $X$. |
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