Factorial moment generating function: Difference between revisions

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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>{{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 {{Bare URL|archive-url=https://web.archive.org/web/20120331042031/https://files.nyu.edu/bpn207/public/Teaching/2005/Stat/Generating_Functions.pdf PDF|archive-date=March2012-03-31 2022|website=nyu.edu}}</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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{{DEFAULTSORT:Factorial Moment Generating Function}}

[[Category:Factorial and binomial topics]]

[[Category:MomentMoments (mathematics)]]

[[Category:Generating functions]]