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wherever this expectation exists. The factorial moment generating function generates the [[factorial moment]]s of the [[probability distribution]].
Provided the factorial moment generating function exists in an interval around ''t'' = 1, the ''n''th factorial moment is given by
::<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)
where the [[Pochhammer symbol]] (''x'')<sub>''n''</sub> is the [[falling factorial]]
:<math>(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,</math>
(Confusingly, some mathematicians, especially in the field of [[special function]]s, use the same notation to represent the [[rising factorial]].)
==Example==
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