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| In [[probability theory]], the '''factorial moment''' is a mathematical quantity defined as the [[Expected value|expectation]] or average of the [[falling factorial]] of a [[random variable]]. Factorial moments are useful for studying [[non-negative]] [[integer]]-valued random variables.<ref name="daleyPPI2003">D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003.</ref> and arise in the use of [[probability-generating function]]s to derive the moments of discrete random variables.
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| Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.<ref>{{cite book|last=Riordan|first=John|authorlink=John Riordan (mathematician)|title=Introduction to Combinatorial Analysis|year=1958|publisher=Dover}}</ref>
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| ==Definition==
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| The ''r''-th factorial moment of a [[probability distribution]], or, in other words, a [[random variable]] ''X'' with that probability distribution, is:<ref>{{cite book|last=Riordan|first=John|authorlink=John Riordan (mathematician)|title=Introduction to Combinatorial Analysis|year=1958|publisher=Dover|pages=30}}</ref>
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| :<math>\operatorname{E}\bigl[(X)_r\bigr] = \operatorname{E}\bigl[ X(X-1)(X-2)\cdots(X-r+1)\bigr] </math>
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| where the ''E'' is the [[Expected value|expectation]] ([[Operator_(mathematics)#Probability_theory|operator]]) and
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| :<math>(x)_r=x(x-1)(x-2)\cdots(x-r+1)</math> | |
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| is the [[falling factorial]], which gives rise to the name, although the notation <math>(x)_r</math> varies depending on the mathematical field. {{efn|Confusingly, this same notation, the [[Pochhammer symbol]] (''x'')<sub>''r''</sub>, is used, especially in the theory of [[special function]]s, to denote the [[rising factorial]] ''x''(''x'' + 1)(''x'' + 2) ... (''x'' + ''r'' − 1);.<ref name="NIST:DLMF">{{cite book| title=NIST Digital Library of Mathematical Functions| url=http://dlmf.nist.gov/| accessdate=9 November 2013}}</ref> whereas the present notation is used more often in [[combinatorics]].}}
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| ==Examples==
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| ===Poisson distribution===
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| If a random variable ''X'' has a [[Poisson distribution]] with parameter or [[expected value]] λ, then the ''r''-th factorial moment of ''X'' is
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| :<math>\operatorname{E}\bigl[(X)_r\bigr] =\lambda^r.</math>
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| The Poisson distribution has a factorial moment with straightforward form compared to [[Poisson_distribution#Higher_moments|its moments]], which involve [[Stirling numbers of the second kind]].
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| ===Binomial distribution===
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| If a random variable ''X'' has a [[Binomial distribution]] with parameters ''p'' and ''n'', then the ''r''-th factorial moment of ''X'' is:<ref name="potts1953note">{{cite journal| author=Potts, RB| title=Note on the factorial moments of standard distributions| journal=Australian Journal of Physics| year=1953| volume=6| number=4| pages=498–499| publisher=CSIRO| accessdate=13 November 2013}}</ref>
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| :<math>\operatorname{E}\bigl[(X)_r\bigr] = \frac{n!}{(n-r)!} p^r, </math>
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| where ''!'' denotes the [[factorial]] of a non-negative integer.
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| ===Hypergeometric distribution===
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| If a random variable ''X'' has a [[hypergeometric distribution]] with parameters ''N'', ''n'', and ''K'', then the ''r''-th factorial moment of ''X'' is:<ref name="potts1953note"/>
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| :<math>\operatorname{E}\bigl[(X)_r\bigr] = \frac{(K)!}{(K-r)!} \frac{n!}{(n-r)!} \frac{(N-r)!}{N!} .</math>
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| ==See also==
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| * [[Factorial moment measure]]
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| * [[Moment (mathematics)]]
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| * [[Cumulant]]
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| * [[Factorial moment generating function]]
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| ==Notes==
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| {{notelist}}
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Factorial Moment}}
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| [[Category:Probability distributions]]
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| [[Category:Factorial and binomial topics]]
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