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]
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{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\operatorname {E} {\bigl [}X(X-1)(X-2)\cdots (X-r+1){\bigr ]},}
![{\displaystyle \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/21967b6d8a0c514ce6ac058c019090f9cdb62d85)
where the E is the expectation (operator) and
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{\displaystyle (x)_{r}:=\underbrace {x(x-1)(x-2)\cdots (x-r+1)} _{r{\text{ factors}}}\equiv {\frac {x!}{(x-r)!}}}
is the falling factorial, which gives rise to the name, although the notation (x)r varies depending on the mathematical field.[a] 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 pr is the probability that any r of the n trials are all successes, then[5]
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{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=n(n-1)(n-2)\cdots (n-r+1)p_{r}}
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=n(n-1)(n-2)\cdots (n-r+1)p_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73701a3d1cada458997259cea762753c6b642b4b)
Examples
Poisson distribution
If a random variable X has a Poisson distribution with parameter λ, then the factorial moments of X are
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{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r},}
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/374202307218b986ee526e682ba9546e658bed0f)
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]
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{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\binom {n}{r}}p^{r}r!=(n)_{r}p^{r},}
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\binom {n}{r}}p^{r}r!=(n)_{r}p^{r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1be2aedfa8f00f15f8dbc79b80ae3c1c6231bb7e)
where by convention,
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{\displaystyle \textstyle {\binom {n}{r}}}
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]
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{\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}}}.}
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/914cdf854b02df007e95e783f8292bbf475a6c43)
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
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{\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 )}}}
![{\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 )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8898e6be4868e804d4bd5b9919e8ea587048424b)
Calculation of moments
The rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula
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{\displaystyle \operatorname {E} [X^{r}]=\sum _{j=1}^{r}\left\{{r \atop j}\right\}\operatorname {E} [(X)_{j}],}
![{\displaystyle \operatorname {E} [X^{r}]=\sum _{j=1}^{r}\left\{{r \atop j}\right\}\operatorname {E} [(X)_{j}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3dcfa584b7d113d31afa011f744ad063545f6c6)
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. p. 30.
- NIST Digital Library of Mathematical Functions. 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.
- Potts, RB (1953). "Note on the factorial moments of standard distributions". Australian Journal of Physics. 6 (4). CSIRO: 498–499. Bibcode:1953AuJPh...6..498P. doi:10.1071/ph530498.