Factorial moment
Template:Short description 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,<ref name="daleyPPI2003">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</ref> 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.<ref>Template:Cite book</ref>
DefinitionEdit
For a natural number Template:Math, the Template:Math-th factorial moment of a probability distribution on the real or complex numbers, or, in other words, a random variable Template:Math with that probability distribution, is<ref>Template:Cite book</ref>
- <math>\operatorname{E}\bigl[(X)_r\bigr] = \operatorname{E}\bigl[ X(X-1)(X-2)\cdots(X-r+1)\bigr],</math>
where the Template:Math is the expectation (operator) and
- <math>(x)_r := \underbrace{x(x-1)(x-2)\cdots(x-r+1)}_{r \text{ factors}} \equiv \frac{x!}{(x-r)!}</math>
is the falling factorial, which gives rise to the name, although the notation Template:Math varies depending on the mathematical field.Template:Efn Of course, the definition requires that the expectation is meaningful, which is the case if Template:Math or Template:Math.
If Template:Math is the number of successes in Template:Math trials, and Template:Math is the probability that any Template:Math of the Template:Math trials are all successes, then<ref>P.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261.</ref>
- <math>\operatorname{E}\bigl[(X)_r\bigr] = n(n-1)(n-2)\cdots(n-r+1)p_r</math>
ExamplesEdit
Poisson distributionEdit
If a random variable Template:Math has a Poisson distribution with parameter λ, then the factorial moments of Template:Math are
- <math>\operatorname{E}\bigl[(X)_r\bigr] =\lambda^r,</math>
which are simple in form compared to its moments, which involve Stirling numbers of the second kind.
Binomial distributionEdit
If a random variable Template:Math has a binomial distribution with success probability Template:MathTemplate:Closed-closed and number of trials Template:Math, then the factorial moments of Template:Math are<ref name="potts1953note">Template:Cite journal</ref>
- <math>\operatorname{E}\bigl[(X)_r\bigr] = \binom{n}{r} p^r r! = (n)_r p^r,</math>
where by convention, <math>\textstyle{\binom{n}{r}} </math> and <math>(n)_r</math> are understood to be zero if r > n.
Hypergeometric distributionEdit
If a random variable Template:Math has a hypergeometric distribution with population size Template:Math, number of success states Template:Math} in the population, and draws Template:Math}, then the factorial moments of Template:Math are <ref name="potts1953note"/>
- <math>\operatorname{E}\bigl[(X)_r\bigr] = \frac{\binom{K}{r}\binom{n}{r}r!}{\binom{N}{r}} = \frac{(K)_r (n)_r}{(N)_r}. </math>
Beta-binomial distributionEdit
If a random variable Template:Math has a beta-binomial distribution with parameters Template:Math, Template:Math, and number of trials Template:Math, then the factorial moments of Template:Math are
- <math>\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)} </math>
Calculation of momentsEdit
The rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula
- <math>\operatorname{E}[X^r] = \sum_{j=1}^r \left\{ {r \atop j} \right\} \operatorname{E}[(X)_j], </math>
where the curly braces denote Stirling numbers of the second kind.