Factorial moment generating function

In probability theory and statistics, the factorial moment generating function 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 ${\displaystyle |t|=1}$, see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then ${\displaystyle M_{X}}$ is also called probability-generating function of X and ${\displaystyle M_{X}(t)}$ is well-defined at least for all t on the closed unit disk ${\displaystyle |t|\leq 1}$.

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