Conditional probability with poisson?

I'm reviewing for a final exam, and I got stuck on one of the practice problems, and was hoping someone could point me in the right direction:

The question is: given $\displaystyle X = U + V, Y = V + W$, where $\displaystyle U, V$ and $\displaystyle W$ are independant Poissons with different means, find $\displaystyle E[Y|X]$

So to do this, I figure I'd need to find $\displaystyle f(y|x) = \frac{f(x,y)}{f(x)}$, but I'm having trouble getting $\displaystyle f(x,y)$.

I have: $\displaystyle f(x,y) = P(U+V=x,V+W=y)$, which I can use independance (of U & W) to separate and get $\displaystyle P(U+V=x,V+W=y) = P(U = x-v, W = y-v) = P(U=x-v)P(W=y-v)$ but I think that would actually be $\displaystyle f(x,y,v)$, and I'm not sure marginalizing over v to get $\displaystyle f(x,y)$ is the right approach.

Could someone tell me if this is the right direction to head in, or if there's a better/easier way to get $\displaystyle f(y|x)$?

thanks!