I have this problem:
Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables. Find Cov($X, Y$).
I tried to use the properties of covariance:
Cov$(X, Y) = Cov(V + W, V + Z) = Cov(V, V) + Cov(V, Z) + Cov(W, V) + Cov(W, Z) = Var(V) + Cov(V + Z) + Cov(W, V) + Cov(W,Z)$
My solutions say that the answer is Var($X$) but I'm not sure how to get it. I'd be thankful for help on how we get Var($X$) as the answer.
Covariance of any two independent random variables is $0$. So you get $var(V)+0+0+0=var(V)$. $var (X)$ is not the correct answer.