Factorial moment generating function: Difference between revisions

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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>

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]]s, then <math>M_X</math> is also called [[probability-generating function]] (PGF) 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>.

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

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

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>{{Cite web |last=Néri |first=Breno de Andrade Pinheiro |date=2005-05-23 |title=Generating Functions |url=http://homepages.nyu.edu/~bpn207/Teaching/2005/Stat/Generating_Functions.pdf |archive-url=https://web.archive.org/web/20120331042031/https://files.nyu.edu/bpn207/public/Teaching/2005/Stat/Generating_Functions.pdf |archive-date=2012-03-31 |website=nyu.edu}}</ref>

::<math>E\left(operatorname{E}[(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]]

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

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

==Examples==

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

===Poisson distribution===

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

:<math>M_X(t)

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

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

</math>

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

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

==See also==

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

* [[momentMoment (mathematics)]]

* [[Moment-generating function]]

* [[Cumulant-generating function]]

==References==

::<math>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>

{{Reflist}}

==Example==

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>

and thus we have

:<math>E( (X)_n )=\lambda^n.</math>

==See also==

* [[Factorial moment]]

* [[moment (mathematics)]]

{{DEFAULTSORT:Factorial Moment Generating Function}}

[[Category:Probability distributions]]

[[Category:Factorial and binomial topics]]

[[Category:Moments (mathematics)]]

[[Category:Generating functions]]