Method for Calculating Higher Moments

Question

During my studies I've come across this "shortcut" for calculating higher moments. How is this method (of calculating moments by summation of a combination of lower moments) called, and how does it work in general?

In particular, I've seen it used for Poisson and Binomial distribution.

Example: E[X³] = E[X(X-1)(X-2)] + 3E[X(X-1)] + E[X]

While generally x(x-1)(x-2) + 3x(x-1) + x = x³, I believe (X-1) above refers to a previous realization of the random variable X. (?)

Example for E[X^3] and E[X^4]

Answer 1

Thanks, the term "Factorial Moment" is a sufficient first pointer. The corresponding Wikipedia article is quite short, but was helpful in combination with this thread:

Calculating factorial moment for distributions