if Y1 and Y2 are independent, does it follow that U and X are independent, where U = f1(Y1, Y2) and X= f2(Y1, Y2)

Question

If X, Y are iid rvs, and U and Z are r.v.s that can each be written in terms of X and Y, does that mean that U and Z are independent?

Answer 1

No. Consider i.i.d. standard normal random variables, $\,X,Y\sim\mathcal{N}\left(0,1\right)\,.$ Let $\,\,U{}={}a_1X+a_2Y\,\,$ and $\,\,Z{}={}b_1X+b_2Y$ for non-zero real numbers $a_1, a_2, b_1$ and $b_2$.

Then, see that

$$ \mathbb{C}ov\left(U,Z\right){}={}a_1b_1+a_2b_2\ne0\,, $$

therefore $U$ and $Z$ can't be independent.

Answer 2

No, and there's a trivial counterexample (or example, depending on how you interpret it): $U=Z=X+Y$ (or any other expression, even a constant, instead of $X$ and $Y$).

Answer 3

Even more trivial: Let $U = Z$ be the same function of $X$ and $Y$. Then $U$ and $Z$ are obviously dependent. Or let $U = g(Z)$ for some non-degenerate function $g$.