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|] < ∞.

If X is the number of successes in n trials, and p_r 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

${\displaystyle \operatorname {E} [X^{r}]=\sum _{j=1}^{r}\left\{{r \atop j}\right\}\operatorname {E} [(X)_{j}],}$

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

See also

• Factorial moment measure
• Moment (mathematics)
• Cumulant
• Factorial moment generating function

Notes

1. The Pochhammer symbol (x)_r is used especially in the theory of special functions, to denote the falling factorial x(x - 1)(x - 2) ... (x - r + 1);.^[4] whereas the present notation is used more often in combinatorics.