Suppose that $X$ is a Poisson random variable with parameter $\lambda$, that is, $$ P(X = k) = \frac{\lambda^k}{k!}e^{-\lambda}, \quad \quad k \in \mathbb{N} $$ For $n \geq 2$, I want to prove that $$ E(X(X-1) \ldots(X-n+1))=\lambda^{n}. $$
I've tried to show by using an induction on $n$.
Let $n=2$. Using the fact that both the expectation $E(X) $ and variance $\sigma_X^2$ are equal to $\lambda$ and $$ \sigma_X^2 = E(X^2) - E(X)^2, $$ I obtained that $E(X^2) = \lambda + \lambda^2$. This gave me that $$ E(X(X-1)) = E(X^2 - X) = \lambda + \lambda^2 - \lambda = \lambda^2. $$ Hence, this made me think that induction might be a good idea to prove. But, I can't complete the induction. Do you have an idea ?