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

for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle , see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then is also called probability-generating function (PGF) of X and is well-defined at least for all t on the closed unit disk .

The factorial moment generating function generates the factorial moments of the probability distribution. Provided exists in a neighbourhood of t = 1, the nth factorial moment is given by [1]

where the Pochhammer symbol (x)n is the falling factorial

(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)

Examples

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Poisson distribution

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Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is

 

(use the definition of the exponential function) and thus we have

 

See also

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References

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  1. ^ Néri, Breno de Andrade Pinheiro (2005-05-23). "Generating Functions" (PDF). nyu.edu. Archived from the original (PDF) on 2012-03-31.