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Factorial Moment Generating Function




:M_X(t)=E\left(t^{X} ight), \quad t \in \mathbb{R},

wherever this expectation exists. The factorial moment generating function generates the Factorial Moment s of the Probability Distribution .

Provided the factorial moment generating function exists in an interval around ''t'' = 1, the ''n''th factorial moment is given by



and thus we have

:E( (X)_n )=\lambda^n.


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