independence between one R.V. and its sum with an independent R.V. If I have two independent, poisson-distributed random variables $X$ and $Y$, and their sum, $ {X+Y}=Z $, is $X$ independent of $Z$? I know that $X$ would be independent of any function of numerous individual random variables of which it is independent, but I'm running into a problem applying this reasoning to a function that involves X itself. 

One simple proof that they are dependent  involves showing that the conditional distribution of Z given X is not the same as the unconditional distribution. That is,
$P(Z \leq z | X=x) = P(x + Y \leq z) = P(Y \leq z-x) \neq P(Z \leq z)$.
