If X and Y are independent random variables and g is a real valued function on the real, how would I show that g(X) and Y are independent?

Question

I understand it in the case where I have to prove g(X) is independent to f(Y) but I am not sure how to go about this example. Thanks.

Possible duplicate of If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent. — lulu, Oct 30 '17 at 16:21
@GabrielRomon You are right (though I don't understand what the question is. How can you understand the general statement here but not the particular?). I retracted my "close vote" and didn't notice that the system left the associated comment in place. I'll leave it there, as of course the answer to that question handles the case when $f$ is the identity. — lulu, Oct 30 '17 at 16:34

score 0 · Answer 1 · answered Nov 01 '17 at 19:48

If $g$ is a measurable function then $\sigma(g(Y))\subset \sigma(Y)$, as for any Borel set $B$, we have $$ g(Y)^{-1}(B) = Y^{-1}(g^{-1}(B))\in\sigma(Y). $$ It follows that if $E\in\sigma(X)$ and $F\in\sigma(g(Y))$, then $f\in\sigma(Y)$ so by independence of $X$ and $Y$, $$ \mathbb P(E\cap F)=\mathbb P(E)\mathbb P(F). $$ Hence, $X$ and $g(Y)$ are independent.

If X and Y are independent random variables and g is a real valued function on the real, how would I show that g(X) and Y are independent?

1 Answers1