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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
for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle , see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then is also called probability-generating function (PGF) of X and is well-defined at least for all t on the closed unit disk .
The factorial moment generating function generates the factorial moments of the probability distribution.
Provided exists in a neighbourhood of t = 1, the nth factorial moment is given by [1]
where the Pochhammer symbol (x)n is the falling factorial
(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)
^Néri, Breno de Andrade Pinheiro (2005-05-23). "Generating Functions" (PDF). nyu.edu. Archived from the original (PDF) on 2012-03-31.
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Global Information
![{\displaystyle M_{X}(t)=\operatorname {E} {\bigl [}t^{X}{\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28b0f116569d2acad64dba16934ba02705748486)
![{\displaystyle \operatorname {E} [(X)_{n}]=M_{X}^{(n)}(1)=\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}\right|_{t=1}M_{X}(t),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c80b95990725adbcd8edad115fcba74f86d8cbd8)
