Calculating $E[Z(Z-1)(Z-2)(Z-3)]$ where $Z$ is Poisson If $Z$ is a variable that distributes by Poisson, with Expected value, $E(Z) = 2.5$. 
I need to solve:
$E[Z(Z-1)(Z-2)(Z-3)]$
So What I thought to do is first:
$E[(Z^2-Z)(Z^2-2Z)(Z^2-3Z)]$
$[E(Z^2)-E(Z)][E(Z^2)-2E(Z)][E(Z^2)-3E(Z)]$
From here I'm not sure how can I insert the $E(Z) = 2.5$ if what I get in the end is $E(Z^2)$

The MGF of the Poisson distribution with parameter $\lambda$ is known.  From that you can get $\mathbb E[Z^k]$ for each $k$ that you need.
