Factorial Moments and Moments of Every Order of the Poisson Distribution

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

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

$$\mathbb{E}\Big[\prod_{q=0}^{p-1}(K-q)\Big]=\mu^{p}.$$

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

$$\mathbb{E}\big[K^{p}\big]\ \le\ (2p)^{p}+2^{p}\,\mu^{p}.$$