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]
![{\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
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|] < ∞.
Examples
Poisson distribution
If a random variable X has a Poisson distribution with parameter or expected value λ ≥ 0, then the factorial moments of X are
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r},\qquad r\in \mathbb {N} _{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b93fe1bb37aa66a38ade812d74622f38aa78daa)
The Poisson distribution has a factorial moments 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 success probability p ∈ [0,1] and number of trials n, then the factorial moments of X are[5]
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}p^{r},\qquad r\in \{0,1,\ldots ,n\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298ba7f8232962d6bc56df19c2c7ba8d2a2de0e2)
where ! denotes the factorial of a non-negative integer. For all r > n, the factorial moments are zero.
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 [5]
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {K!}{(K-r)!}}{\frac {n!}{(n-r)!}}{\frac {(N-r)!}{N!}},\qquad r\in \{0,1,\ldots ,\min\{n,K\}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40cb65cb2e4103e7449144b740e8bf705e4c5ae0)
For all larger r, the factorial moments are zero.
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
![{\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}{\frac {B(\alpha +r,\beta )}{B(\alpha ,\beta )}},\qquad r\in \{0,1,\ldots ,n\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f882e795088a3fc4132c701133777a4843c139c)
where B denotes the beta function. For all r > n, the factorial moments are zero.
Calculation of moments
In the examples above, the n-th moment of the random variable X can be calculated by the formula
![{\displaystyle \operatorname {E} {\bigl [}X^{n}{\bigr ]}=\sum _{r=0}^{n}{\biggl \{}{n \atop r}{\biggr \}}\operatorname {E} {\bigl [}(X)_{r}{\bigr ]},\qquad n\in \mathbb {N} _{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30f6742adc0826e3570cc63bc5b509c69ca7c9b8)
where the curly braces denote Stirling numbers of the second kind.
See also
Notes
- ^ Confusingly, this same notation, the Pochhammer symbol (x)r, is used, especially in the theory of special functions, to denote the rising 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.
- ^ a b Potts, RB (1953). "Note on the factorial moments of standard distributions". Australian Journal of Physics. 6 (4). CSIRO: 498–499. doi:10.1071/ph530498.
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