Factorial moment generating function: Difference between revisions

Revision as of 13:51, 5 May 2011

Revision as of 21:56, 7 May 2011

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{{Unreferenced|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>

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]] 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>.

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</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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* [[Moment-generating function]]

* [[Cumulant-generating function]]

{{Reflist}}

{{DEFAULTSORT:Factorial Moment Generating Function}}