# Factorial moment

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

For a natural number *r*, the *r*-th factorial moment of a probability distribution on the real or complex numbers, or, in other words, a random variable *X* with that probability distribution, is^{[3]}

- [math]\displaystyle{ \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 expectation (operator) and

- [math]\displaystyle{ (x)_r := \underbrace{x(x-1)(x-2)\cdots(x-r+1)}_{r \text{ factors}} \equiv \frac{x!}{(x-r)!} }[/math]

is the falling factorial, which gives rise to the name, although the notation (*x*)_{r} varies depending on the mathematical field. ^{[lower-alpha 1]} Of course, the definition requires that the expectation is meaningful, which is the case if (*X*)_{r} ≥ 0 or E[|(*X*)_{r}|] < ∞.

If X is the number of successes in n trials, and p_{r} is the probability that any r of the n trials are all successes, then^{[5]}

- [math]\displaystyle{ \operatorname{E}\bigl[(X)_r\bigr] = n(n-1)(n-2)\cdots(n-r+1)p_r }[/math]

## Examples

### Poisson distribution

If a random variable *X* has a Poisson distribution with parameter *λ*, then the factorial moments of *X* are

- [math]\displaystyle{ \operatorname{E}\bigl[(X)_r\bigr] =\lambda^r, }[/math]

which are simple in 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 success probability *p* ∈ [0,1] and number of trials *n*, then the factorial moments of *X* are^{[6]}

- [math]\displaystyle{ \operatorname{E}\bigl[(X)_r\bigr] = \binom{n}{r} p^r r! = (n)_r p^r, }[/math]

where by convention, [math]\displaystyle{ \textstyle{\binom{n}{r}} }[/math] and [math]\displaystyle{ (n)_r }[/math] are understood to be zero if *r* > *n*.

### Hypergeometric distribution

If a random variable *X* has a hypergeometric distribution with population size *N*, number of success states *K* ∈ {0,...,*N*} in the population, and draws *n* ∈ {0,...,*N*}, then the factorial moments of *X* are ^{[6]}

- [math]\displaystyle{ \operatorname{E}\bigl[(X)_r\bigr] = \frac{\binom{K}{r}\binom{n}{r}r!}{\binom{N}{r}} = \frac{(K)_r (n)_r}{(N)_r}. }[/math]

### Beta-binomial distribution

If a random variable *X* has a beta-binomial distribution with parameters *α* > 0, *β* > 0, and number of trials *n*, then the factorial moments of *X* are

- [math]\displaystyle{ \operatorname{E}\bigl[(X)_r\bigr] = \binom{n}{r}\frac{B(\alpha+r,\beta)r!}{B(\alpha,\beta)} = (n)_r \frac{B(\alpha+r,\beta)}{B(\alpha,\beta)} }[/math]

## Calculation of moments

The *r*th raw moment of a random variable *X* can be expressed in terms of its factorial moments by the formula

- [math]\displaystyle{ \operatorname{E}[X^r] = \sum_{j=0}^r \left\{ {r \atop j} \right\} \operatorname{E}[(X)_j], }[/math]

where the curly braces denote Stirling numbers of the second kind.

## See also

## Notes

- ↑ The Pochhammer symbol (
*x*)_{r}is used especially in the theory of special functions, to denote the falling factorial*x*(*x*- 1)(*x*- 2) ... (*x*-*r*+ 1);.^{[4]}whereas the present notation is used more often in combinatorics.

## References

- ↑ 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 - ↑ Riordan, John (1958).
*Introduction to Combinatorial Analysis*. Dover. - ↑ Riordan, John (1958).
*Introduction to Combinatorial Analysis*. Dover. pp. 30. - ↑
*NIST Digital Library of Mathematical Functions*. http://dlmf.nist.gov/. Retrieved 9 November 2013. - ↑ P.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261.
- ↑
^{6.0}^{6.1}Potts, RB (1953). "Note on the factorial moments of standard distributions".*Australian Journal of Physics*(CSIRO)**6**(4): 498–499. doi:10.1071/ph530498.

Original source: https://en.wikipedia.org/wiki/Factorial moment.
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