Let X be a discrete random variable with Poisson distribution with parameter $\displaystyle \lambda$

1) Show that:

$\displaystyle E(X^2)=\sum_{k=1}^{\infty} \frac{ke^{-\lambda} \lambda^k}{(k-1)!}$

2) By changing the summation variable, show that $\displaystyle E(X^2)=\lambda E(X)+\lambda$

3) Using (2), show that $\displaystyle Var(X)=\lambda$