# 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]

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

where the E is the expectation (operator) and

${\displaystyle (x)_{r}=x(x-1)(x-2)\cdots (x-r+1)}$

is the falling factorial, which gives rise to the name, although the notation (x)r varies depending on the mathematical field. Template:Efn 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

${\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}=\lambda ^{r},\qquad r\in \mathbb {N} _{0}.}$

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 pTemplate:Closed-closed and number of trails n, then the factorial moments of X are[4]

${\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}p^{r},\qquad r\in \{0,1,\ldots ,n\},}$

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 [4]

${\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {K!}{(K-r)!}}{\frac {n!}{(n-r)!}}{\frac {(N-r)!}{N!}},\qquad r\in \{0,1,\ldots ,\min\{n,K\}\}.}$

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 trails n, then the factorial moments of X are

${\displaystyle \operatorname {E} {\bigl [}(X)_{r}{\bigr ]}={\frac {n!}{(n-r)!}}{\frac {B(\alpha +r,\beta )}{B(\alpha ,\beta )}},\qquad r\in \{0,1,\ldots ,n\},}$

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

${\displaystyle \operatorname {E} {\bigl [}X^{n}{\bigr ]}=\sum _{r=0}^{n}{\biggl \{}{n \atop r}{\biggr \}}\operatorname {E} {\bigl [}(X)_{r}{\bigr ]},\qquad n\in \mathbb {N} _{0},}$

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