# 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. [lower-alpha 1] 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]

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

## 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[6]

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

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 [6]

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

### 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 rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula

$\displaystyle{ \operatorname{E}[X^r] = \sum_{j=0}^r \left\{ {r \atop j} \right\} \operatorname{E}[(X)_j], }$

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