1. ## Poisson Expectation

Suppose X has a Poisson distribution with mean 3. then how to compute E[1/((X+2)(X+1))]?

2. Where are you stuck?

$\displaystyle E\bigg[\dfrac{1}{(x+2)(x+1)}\bigg] =\displaystyle \sum_{x=0}^{\infty} \dfrac{1}{(x+2)(x+1)} \times \dfrac{e^{-\lambda}{\lambda}^x}{x!}$

$\displaystyle =\displaystyle \sum_{x=0}^{\infty} \dfrac{e^{-\lambda}{\lambda}^x}{(x+2)!}$

$\displaystyle = \displaystyle \sum_{x=0}^{\infty} \dfrac{e^{-\lambda}{\lambda}^{(x+2)} \; {\lambda}^{-2}}{(x+2)!}\;=\;......$