moment generating functionMomentGeneratingFunction2001-10-26 02:53:102006-09-18 07:12:45Definition\usepackage{amssymb}
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Given a random variable $X$, the \emph{moment generating function} of $X$ is the following function:\\
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$M_X(t) = E[e^{tX}]$ for $t \in R$ (if the expectation converges).
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It can be shown that if the moment generating function of $X$ is defined on an interval around the origin, then\\
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$E[X^k] = M_X^{(k)}(t) |_{t=0} $\\
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In other words, the $k$th-derivative of the moment generating function evaluated at zero is the $k$th moment of $X$.