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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'' &minus; 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 19:58, 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]

E⁡[(X)r]=E⁡[X(X−1)(X−2)⋯(X−r+1)]

where the E is the expectation (operator) and

(x)r=x(x−1)(x−2)⋯(x−r+1)

is the falling factorial, which gives rise to the name, although the notation (x)r 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

E⁡[(X)r]=λr.

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]

E⁡[(X)r]=n!(n−r)!pr,

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]

E⁡[(X)r]=(K)!(K−r)!n!(n−r)!(N−r)!N!.

See also

Notes

Template:Notelist

References

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  1. ↑ 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.
  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  3. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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