Factorial moment generating function: Difference between revisions

Revision as of 21:56, 7 May 2011

Revision as of 03:31, 24 January 2012

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{{UnreferencedRefimprove|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>

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:<math>\operatorname{E}[(X)_n]=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]]

:<math>(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,</math>

(Confusingly, some mathematicians, especially in the field of [[special function]]s, use the same notation to represent the [[rising factorial]].)

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{{DEFAULTSORT:Factorial Moment Generating Function}}

[[Category:Factorial and binomial topics]]

[[Category:Theory of probability distributions]]