# 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]}

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. Template:Efn Of course, the definition requires that the expectation is meaningful, which is the case if (*X*)_{r} ≥ 0 or E[|(*X*)_{r}|] < ∞.

## Examples

### Poisson distribution

If a random variable *X* has a Poisson distribution with parameter or expected value λ ≥ 0, then the factorial moments of *X* are

The Poisson distribution has a factorial moments with straightforward 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* ∈ Template:Closed-closed and number of trails *n*, then the factorial moments of *X* are^{[4]}

where ! denotes the factorial of a non-negative integer. For all *r* > *n*, the factorial moments are zero.

### 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 ^{[4]}

For all larger *r*, the factorial moments are zero.

### Beta-binomial distribution

If a random variable *X* has a beta-binomial distribution with parameters *α* > 0, *β* > 0, and number of trails *n*, then the factorial moments of *X* are

where *B* denotes the beta function. For all *r* > *n*, the factorial moments are zero.

## Calculation of moments

In the examples above, the *n*-th moment of the random variable *X* can be calculated by the formula

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

## See also

## Notes

## References

- ↑ D. J. Daley and D. Vere-Jones.
*An introduction to the theory of point processes. Vol. I*. Probability and its Applications (New York). Springer, New York, second edition, 2003. - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
^{4.0}^{4.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }}