Conditional Joint Function

Question

Let V,W,Z ~ Poisson(λ) are independent variables and we have:

X = V + W

Y = V + Z

Are Y and X independent under condition V?

I tried to solve this problem with joint distribution but I can't.

Answer 1

I tried to solve this problem with joint distribution but I can't.

Indeed, that's not the way to go.

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You know that $V,W,$ and $Z$ are mutually independent and identically Poisson distributed, and $X=V+W$, $Y=V+Z$.

To demonstrate conditional independence of $X$ and $Y$ given $V$, you must show that: $$\mathsf P(X=x, Y=y\mid V=v) ~~=~~ \mathsf P(X=x\mid V=v)\,\mathsf P(Y=y\mid V=v)$$