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]

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

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. ^{[lower-alpha 1]} 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]

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

Examples

Poisson distribution

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

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

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

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

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

Calculation of moments

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

${\displaystyle \operatorname {E} [X^{r}]=\sum _{j=1}^{r}\left\{{r \atop j}\right\}\operatorname {E} [(X)_{j}],}$

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

See also

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

Notes

1. The Pochhammer symbol (x)_r is used especially in the theory of special functions, to denote the falling factorial x(x - 1)(x - 2) ... (x - r + 1);. ^[4] whereas the present notation is used more often in combinatorics.

References

1. 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
2. Riordan, John (1958). Introduction to Combinatorial Analysis. Dover.
3. Riordan, John (1958). Introduction to Combinatorial Analysis. Dover. p. 30.
4. NIST Digital Library of Mathematical Functions . Retrieved 9 November 2013.
5. P.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261.
6. Potts, RB (1953). "Note on the factorial moments of standard distributions". Australian Journal of Physics. 6 (4). CSIRO: 498–499. Bibcode:1953AuJPh...6..498P. doi: 10.1071/ph530498 .