This is question 3.8.4 of An Introduction to Mathematical Statistics and Its Applications, 5th Edition, by Larsen and Marx. This is not homework for a class I am taking now, but might someday be for a class I'll take in the future.
The goal could be expressed in terms of probabilities, to try and show that for every interval A and B, P(V $\in$ A and X+Y $\in$ B) = P(V $\in$ A) $\centerdot$ P(X+Y \in B).
But, I've been trying to answer the question in terms of pdf's -- to show that f$_{V,X+Y}$(v,w) = f$_V$(v) $\centerdot$ f$_{X+Y}$(w), for W = X + Y, without success.
The other thing I was considering was the equivalence of f$_{V+(X+Y)}$(w), f$_{(V+X)+Y)}$(w), and f$_{(V+Y)+X}$(w), for W = X + Y + V.
Is this the wrong tack? Is there some easy way of proving this, say, using graph theory, with random variables as nodes and dependency relationships as edges?
Thanks in advance for any hints / help.