Factorial moment - Alchetron, The Free Social Encyclopedia

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, and arise in the use of probability-generating functions to derive the moments of discrete random variables.

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

E ⁡ [ ( X ) r ] = E ⁡ [ X ( X − 1 ) ( X − 2 ) ⋯ ( X − r + 1 ) ] ,

where the E is the expectation (operator) and

( x ) r := x ( x − 1 ) ( x − 2 ) ⋯ ( x − r + 1 ) ⏟ r factors ≡ x ! ( x − r ) !

is the falling factorial, which gives rise to the name, although the notation (x)_r varies depending on the mathematical field. Of course, the definition requires that the expectation is meaningful, which is the case if (X)_r ≥ 0 or E[|(X)_r|] < ∞.

Poisson distribution

If a random variable X has a Poisson distribution with parameter λ, then the factorial moments of X are

E ⁡ [ ( X ) r ] = λ 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

E ⁡ [ ( X ) r ] = ( n r ) p r r ! = ( n ) r p r ,

where by convention, ( n r ) and ( 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

E ⁡ [ ( X ) r ] = ( K r ) ( n r ) r ! ( N r ) = ( 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

E ⁡ [ ( X ) r ] = ( n r ) B ( α + r , β ) r ! B ( α , β ) = ( n ) r B ( α + r , β ) B ( α , β )

Calculation of moments

The nth moment of a random variable X can be expressed in terms of its factorial moments by the formula

E ⁡ [ X n ] = ∑ r = 0 n { n k } E ⁡ [ ( X ) r ] ,

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

References

Factorial moment Wikipedia

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