Expected Value Composition

Question

I am given the following:

Let X be Poisson with parameter λ > 0. For any positive integer k, calculate E(X(X − 1)(X − 2) × · · · × (X − k).

I know the answer is as follows: $$\sum _{n >= 0} n(n-1)(n-2) ... λ^k e^{-λ}/n! $$

I'm a little unsure how they were able to come up with this answer from the problem statement. I was thinking it came from the property that expected value is a linear, but am really not sure. Thanks!

Answer 1

That is just by the definition of expectation

$$E[g(X)] = \sum_{x=0}^\infty Pr(X=x) g(x) $$

here $g(x) = \prod_{i=0}^k (x-i)$.

\begin{align} E \left[ \prod_{i=0}^k (X-i) \right] &= \sum_{x=0}^\infty\frac{\exp(-\lambda)\lambda^x}{x!}\prod_{i=0}^k (x-i) \end{align}