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]
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. [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 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, and 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.
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]
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. [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 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, and 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.