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Factorial moment generating function

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In probability theory and statistics, the factorial moment generating function of the probability distribution of a real-valued random variable X is defined as

M X ( t ) = E [ t X ]

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

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

E [ ( X ) n ] = M X ( n ) ( 1 ) = d n d t n | t = 1 M X ( t ) ,

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

( x ) n = x ( x 1 ) ( x 2 ) ( x n + 1 ) .

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

Example

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

M X ( t ) = k = 0 t k P ( X = k ) = λ k e λ / k ! = e λ k = 0 ( t λ ) k k ! = e λ ( t 1 ) , t C ,

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

E [ ( X ) n ] = λ n .

References

Factorial moment generating function Wikipedia