In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the nth factorial moment of a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

, also called the nth factorial moment of any random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 X with that probability distribution, is


where


is the falling factorial (confusingly, this same notation, the Pochhammer symbol

Pochhammer symbol

In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 (x)_n, is used by some mathematicians, especially in the theory of special functions, to denote the rising factorial x(x + 1)(x + 2) ... (x + n − 1); the present notation is used by combinatorialists

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

).

For example, if X has a Poisson distribution

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 and/or space if these events occur with a known average rate and independently of the time since...

 with expected value

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 λ, then the nth factorial moment of X is


One context in which factorial moments occur naturally is at an initial stage in the use of probability-generating function

Probability-generating function

In probability theory, the probability-generating function of a discrete random variable is a power series representation of the probability mass function of the random variable...

s to derive the moments of discrete distributions.