I have this problem:
Let $X = V + W$ and $Y = V + Z$ where $V, W, Z$ are independent Pois($\lambda$) random variables.
I found that $Cov(X, Y) = Var(V) = \lambda$
It now asks to find whether $X$ and $Y$ are conditionally independent given $V$.
I am now trying to find if $X$ and $Y$ are conditionally independent given $V$.
So I start with $P(X = x, Y = y | V = v) =$
Now, I know that, for conditional independence, I have to show that the joint probability mass function factors into the product of its marginal probability mass functions.
But where do I go from here? This is where I have been stuck.
Please show me how this is done. Thank you very much for your help!