Factorial moment generating function

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In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as

:$M\_X(t)=\backslash operatorname\{E\}\backslash bigl[t^\{X\}\backslash bigr]$

for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle $|t|=1$, see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then $M\_X$ is also called probability-generating function (PGF) of X and $M\_X(t)$ is well-defined at least for all t on the closed unit disk $|t|\backslash le1$.

The factorial moment generating function generates the factorial moments of the probability distribution.

Provided $M\_X$ exists in a neighbourhood of t = 1, the nth factorial moment is given by {{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}}

:$\backslash operatorname\{E\}[(X)\_n]=M\_X^\{(n)\}(1)=\backslash left.\backslash frac\{\backslash mathrm\{d\}^n\}\{\backslash mathrm\{d\}t^n\}\backslash right|\_\{t=1\}\; M\_X(t),$

where the Pochhammer symbol (x)_n is the falling factorial

:$(x)\_n\; =\; x(x-1)(x-2)\backslash cdots(x-n+1).\backslash ,$

(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)