Factorial Moment
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the factorial moment is a mathematical quantity defined as the expectation or average of the
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. Factorial moments are useful for studying
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
-valued random variables,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 and arise in the use of
probability-generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are oft ...
s 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.


Definition

For a natural number , the -th factorial moment of a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
on the real or complex numbers, or, in other words, a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with that probability distribution, is :\operatorname\bigl X)_r\bigr= \operatorname\bigl X(X-1)(X-2)\cdots(X-r+1)\bigr where the is the expectation ( operator) and :(x)_r := \underbrace_ \equiv \frac is the
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
, which gives rise to the name, although the notation varies depending on the mathematical field. Of course, the definition requires that the expectation is meaningful, which is the case if or . If is the number of successes in trials, and is the probability that any of the trials are all successes, thenP.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261. :\operatorname\bigl X)_r\bigr= n(n-1)(n-2)\cdots(n-r+1)p_r


Examples


Poisson distribution

If a random variable has a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
with parameter ''λ'', then the factorial moments of are :\operatorname\bigl X)_r\bigr=\lambda^r, which are simple in form compared to its moments, which involve
Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
.


Binomial distribution

If a random variable has a
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
with success probability and number of trials , then the factorial moments of are :\operatorname\bigl X)_r\bigr= \binom p^r r! = (n)_r p^r, where by convention, \textstyle and (n)_r are understood to be zero if ''r'' > ''n''.


Hypergeometric distribution

If a random variable has a
hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without'' ...
with population size , number of success states in the population, and draws , then the factorial moments of are :\operatorname\bigl X)_r\bigr= \frac = \frac.


Beta-binomial distribution

If a random variable has a
beta-binomial distribution In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of B ...
with parameters , , and number of trials , then the factorial moments of are :\operatorname\bigl X)_r\bigr= \binom\frac = (n)_r \frac


Calculation of moments

The ''r''th raw moment of a random variable ''X'' can be expressed in terms of its factorial moments by the formula :\operatorname ^r= \sum_^r \left\ \operatorname X)_j where the curly braces denote
Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
.


See also

* Factorial moment measure *
Moment (mathematics) In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mas ...
*
Cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
*
Factorial moment generating function In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable ''X'' is defined as :M_X(t)=\operatorname\bigl ^\bigr/math> for all complex numbers ''t'' for w ...


Notes


References

{{DEFAULTSORT:Factorial Moment Moment (mathematics) Factorial and binomial topics