Factorial moment

The term "factorial moment" refers to a specific type of moment used in probability theory and statistics. Factorial moments are particularly useful when dealing with discrete random variables, especially in the context of counting and combinatorial problems. For a discrete random variable \( X \) taking non-negative integer values, the \( n \)-th factorial moment is defined as: \[ E[X^{(n)}] = E\left[\frac{X!}{(X-n)!