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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'' = 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
:<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'')<sub>''n''</sub> is the [[falling factorial]]
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