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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. | |||
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> | |||
==Definition== | |||
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> | |||
:<math>\operatorname{E}\bigl[(X)_r\bigr] = \operatorname{E}\bigl[ X(X-1)(X-2)\cdots(X-r+1)\bigr] </math> | |||
where the ''E'' is the [[Expected value|expectation]] ([[Operator_(mathematics)#Probability_theory|operator]]) and | |||
:<math>(x)_r=x(x-1)(x-2)\cdots(x-r+1)</math> | |||
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]].}} | |||
==Examples== | |||
===Poisson distribution=== | |||
If a random variable ''X'' has a [[Poisson distribution]] with parameter or [[expected value]] λ, then the ''r''-th factorial moment of ''X'' is | |||
:<math>\operatorname{E}\bigl[(X)_r\bigr] =\lambda^r.</math> | |||
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]]. | |||
===Binomial distribution=== | |||
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> | |||
:<math>\operatorname{E}\bigl[(X)_r\bigr] = \frac{n!}{(n-r)!} p^r, </math> | |||
where ''!'' denotes the [[factorial]] of a non-negative integer. | |||
===Hypergeometric distribution=== | |||
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"/> | |||
:<math>\operatorname{E}\bigl[(X)_r\bigr] = \frac{(K)!}{(K-r)!} \frac{n!}{(n-r)!} \frac{(N-r)!}{N!} .</math> | |||
==See also== | |||
* [[Factorial moment measure]] | |||
* [[Moment (mathematics)]] | |||
* [[Cumulant]] | |||
* [[Factorial moment generating function]] | |||
==Notes== | |||
{{notelist}} | |||
==References== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Factorial Moment}} | |||
[[Category:Probability distributions]] | |||
[[Category:Factorial and binomial topics]] | |||
Revision as of 20:58, 6 May 2013
In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables.[1] and arise in the use of probability-generating functions to derive the moments of discrete random variables.
Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.[2]
Definition
The r-th factorial moment of a probability distribution, or, in other words, a random variable X with that probability distribution, is:[3]
![\operatorname {E}{\bigl [}(X)_{r}{\bigr ]}=\operatorname {E}{\bigl [}X(X-1)(X-2)\cdots (X-r+1){\bigr ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72d54569b78fd4f62089d26f2f4542f7095b09bc)
where the E is the expectation (operator) and
is the falling factorial, which gives rise to the name, although the notation varies depending on the mathematical field. Template:Efn
Examples
Poisson distribution
If a random variable X has a Poisson distribution with parameter or expected value λ, then the r-th factorial moment of X is
![\operatorname {E}{\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/b797bc869961f4cfaa20038c331c3ba652692ba5)
The Poisson distribution has a factorial moment with straightforward form compared to its moments, which involve Stirling numbers of the second kind.
Binomial distribution
If a random variable X has a Binomial distribution with parameters p and n, then the r-th factorial moment of X is:[4]
![\operatorname {E}{\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}p^{r},](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9c5b886ef0508fd0d48172e06ad4dc142e0f94)
where ! denotes the factorial of a non-negative integer.
Hypergeometric distribution
If a random variable X has a hypergeometric distribution with parameters N, n, and K, then the r-th factorial moment of X is:[4]
![\operatorname {E}{\bigl [}(X)_{r}{\bigr ]}={\frac {(K)!}{(K-r)!}}{\frac {n!}{(n-r)!}}{\frac {(N-r)!}{N!}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/432627578897468a37da4e62ac5c1512095ef553)
See also
Notes
References
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- ↑ 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.
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