# 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

${\displaystyle (x)_{r}:=\underbrace {x(x-1)(x-2)\cdots (x-r+1)} _{r{\text{ factors}}}\equiv {\frac {x!}{(x-r)!}}}$

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|] < ∞.

## Examples

### Poisson distribution

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

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

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 [0,1] and number of trials n, then the factorial moments of X are[5]

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

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

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

### 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

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

## Calculation of moments

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

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

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