If $X$, $Y$ and $Z$ are mutually independent random variables, is it true that $X+Y (X \cdot Y,X/Y,\dots)$ and $Z$ are independent?

Question

I just wonder that given that RVs $X, Y, Z$ are mutually independent, how should I quickly determine whether a combination of $X$ and $Y$, e.g. $X+Y, X\cdot Y, X/Y, X^Y$, is independent of $Z$?

Are there some general conclusions?

Thanks in advance.

Answer 1

For any measurable function $f: \mathbb R^{2} \to \mathbb R$, $Z$ is independent of $f(X,Y)$. Hence the answer is YES in all these cases (assuming that $X/Y$ and $X^{Y}$ are defined).

Proof: $$P(f(X,Y) \in E, Z \in F)=P((X,Y) \in f^{-1}(E), Z \in F)$$ $$=P((X,Y) \in f^{-1}(E))P( Z \in F)$$ $$=P(f(X,Y) \in E) P( Z \in F)$$ for any Borel sets $E$ and $F$ in $\mathbb R$.

The last equality is proved by considering the collection of all Borel set $D$ in $\mathbb R^{2}$ such that $P((X,Y) \in D, Z \in C)=P((X,Y) \in D)P(Z \in C)$, and verifying that this is a sigma algebra which contains sets of the form $A \times B$ where $A$ and $B$ are Borel sets in $\mathbb R$.