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]

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

Poisson distribution edit

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 edit

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 edit

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 edit

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

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

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.

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