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.
[2] 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] where the E is the expectation (operator) and is the falling factorial, which gives rise to the name, although the notation (x)r varies depending on the mathematical field.
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] If a random variable X has a Poisson distribution with parameter λ, then the factorial moments of X are which are simple in form compared to its moments, which involve Stirling numbers of the second kind.
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] where by convention,
are understood to be zero if r > n. 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] 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 The rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula where the curly braces denote Stirling numbers of the second kind.