Factorial moment generating function: Difference between revisions

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Revision as of 04:26, 17 December 2009

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{{Unreferenced|date=December 2009}}

In [[probability theory]] and [[statistics]], the '''factorial moment generating function''' of the [[probability distribution]] of a [[real number|real-valued]] [[random variable]] ''X'' is defined as

:<math>M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]</math>

for all [[complex number]]s ''t'' for which this [[expected value]] exists. This is the case at least for all ''t'' on the [[unit circle]] <math>|t|=1</math>, see [[characteristic function (probability theory)|characteristic function]]. If&nbsp;''X'' is a discrete random variable taking values only in the set {0,1, ...} of non-negative [[integer|integers]]s, then <math>M_X</math> is also called [[probability-generating function]] of ''X'' and <math>M_X(t)</math> is well-defined at least for all ''t'' on the [[closed set|closed]] [[unit disk]] <math>|t|\le1</math>.

The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].

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==Example==

Suppose ''X'' has a [[Poisson distribution]] with [[expected value]] &lambda;λ, then its factorial moment generating function is

:<math>M_X(t)

=\sum_{k=0}^\infty t^k\underbrace{\operatorname{P}(X=k)}_{=\,\lambda^ke^{-\lambda}/k!}

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* [[Cumulant-generating function]]

{{DEFAULTSORT:Factorial Moment Generating Function}}

[[Category:Probability theory]]

[[Category:Factorial and binomial topics]]

[[Category:Articles lacking sources (Erik9bot)]]

[[Category:Theory of probability distributions]]