1. ## Poisson distribution

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

1) Show that:

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

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

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

2. Originally Posted by acevipa
Let X be a discrete random variable with Poisson distribution with parameter $\lambda$

1) Show that:

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

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

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