poisson distribution find exact sum

Question

Using only properties of Poisson distribution find the exact value of this sum

$\sum_{x=0}^{\infty}\frac{x^22^x}{x!}$

I believe $\lambda$ = 2
(E(X))$^2$ = 4
V(X) = 2
E(X$^2$) = 6

I don't know how to find the exact value of the sum.

Answer 1

Let $X$ have Poisson distribution with parameter $\lambda=2$. You showed that $E(X^2)=6$. Note that $$6=E(X^2)=\sum_0^\infty x^2e^{-2}\frac{2^x}{x!}.$$ Multiply both sides by $e^2$. We conclude that our sum is $6e^2$.