Factorial moment generating function: Difference between revisions

Revision as of 17:59, 2 December 2006

Revision as of 20:32, 13 July 2007

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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)=\operatorname{E}\left(bigl[t^{X}\right)bigr], \quad t \in \mathbb{R},</math>

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

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Provided the factorial moment generating function exists in an interval around ''t''&nbsp;=&nbsp;1, the ''n''th factorial moment is given by

:<math>\operatorname{E\left(}[(X)_n\right)]=M_X^{(n)}(1)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=1} M_X(t),</math>

where the [[Pochhammer symbol]] (''x'')_''n'' is the [[falling factorial]]

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==Example==

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

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

(use the [[Exponential_function#Formal_definition|definition of the exponential function]]) and thus we have

and thus we have

:<math>\operatorname{E( }[(X)_n )]=\lambda^n.</math>

==See also==

* [[Factorial moment]]

* [[moment (mathematics)]]

[[Category:Probability distributionstheory]]

[[Category:Factorial and binomial topics]]