Factorial moment generating function: Difference between revisions

Revision as of 15:05, 14 March 2006

Revision as of 21:45, 14 March 2006

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In [[probability theory]] and [[statistics]], the '''factorial moment generating function''' of the [[probability distribution]] of a [[random variable]] ''X'' is

:<math>M_X(t)=E\left(t^{X}\right), \quad t \in \mathbb{R},</math>

wherever this expectation exists. The factorial moment generating function generates the [[factorial moment|factorial moments]] of the [[probability distribution]].

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

:<math>M_X(t) = \sum_{x=0}^\infty \frac{(t\lambda)^x e^{-\lambda}}{x!} = e^{-\lambda(1-t)},</math>

:::<math> = e^{-\lambda(1-t)}</math>

and thus we have