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]

\operatorname {E}{\bigl [}(X)_{r}{\bigr ]}=\operatorname {E}{\bigl [}X(X-1)(X-2)\cdots (X-r+1){\bigr ]},

where the $E$ is the expectation (operator) and

(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. ^{[lower-alpha 1]} Of course, the definition requires that the expectation is meaningful, which is the case if $(X) r \geq 0$ or $E[|(X) r |] < \infty$ .

Examples

Poisson distribution

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

\operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r},

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 $p \in$ $[0,1]$ and number of trials $n$ , then the factorial moments of $X$ are^[5]

\operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\binom {n}{r}}p^{r}r!=(n)_{r}p^{r},

where by convention, $\textstyle {\binom {n}{r}}$ and $(n)_{r}$ 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 $K \in {0,..., N$ } in the population, and draws $n \in {0,..., N$ }, then the factorial moments of $X$ are ^[5]

\operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {{\binom {K}{r}}{\binom {n}{r}}r!}{\binom {N}{r}}}={\frac {(K)_{r}(n)_{r}}{(N)_{r}}}.

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

\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 )}}

Calculation of moments

The nth moment of a random variable X can be expressed in terms of its factorial moments by the formula

\operatorname {E} [X^{n}]=\sum _{r=0}^{n}\left\{{n \atop k}\right\}\operatorname {E} [(X)_{r}],

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

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.
1 2 Potts, RB (1953). "Note on the factorial moments of standard distributions". Australian Journal of Physics. CSIRO. 6 (4): 498–499. doi:10.1071/ph530498.

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