Factorial Moments and Moments of Every Order of the Poisson Distribution

Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mu\ge0$ be a \reftext{def:real-numbers-c54-2026c}{real number}, let $K$ be a random variable on $(\Omega,\mathcal{F},P)$ with the \reftext{def:poisson-distribution-2026b}{Poisson distribution} with parameter $\mu$, and let $p$ be a \reftext{def:natural-numbers-2026a}{natural number}. Write $\mathbb{E}$ for the \reftext{def:expectation-variance-2026a}{expectation} and use the \reftext{def:finite-product-notation-2026a}{finite product notation}. Then:

\textbf{(a) (Factorial moments.)} The random variable $\prod_{q=0}^{p-1}(K-q)=K(K-1)\cdots(K-p+1)$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable}, and

\textbf{(b) (Moments of every order.)} The random variable $K^{p}$ is integrable, and