Factorial moment generating function: Difference between revisions

Revision as of 15:14, 14 June 2021

Revision as of 06:31, 13 March 2022

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{{RefimproveMore citations needed|date=December 2009}}

In [[probability theory]] and [[statistics]], the '''factorial moment generating function''' (FMGF) 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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The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].

Provided <math>M_X</math> exists in a [[neighbourhood (mathematics)|neighbourhood]] of ''t''&nbsp;=&nbsp;1, the ''n''th factorial moment is given by <ref>http://homepages.nyu.edu/~bpn207/Teaching/2005/Stat/Generating_Functions.pdf {{Bare URL PDF|date=March 2022}}</ref>

:<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]]

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=e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},

</math>

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

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