TITLE: If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$?
QUESTION [7 upvotes]: If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$?
I always have confusion about it.
I feel ... yeah of course $f(X)$ and $g(Y)$ are independent, because $X$ and $Y$ are.
But .. is it right?? Then how can I prove it?

REPLY [11 votes]: By definition we have that $X,Y$ are independent if $F(x,y) = F_X (x) F_Y(y)$. That is that
$$     P\{X \le x , Y \le y \} = P \{X \le x \} P \{Y \le y \} $$
with this you can proof that, for any $A,B$ Borel sets
$$     P\{X \in A , Y \in B \} = P \{X \in A \} P \{Y \in B \} $$
holds. Now let $g,h$ be measurable functions
\begin{align*}
     F_{g(X), g(Y)} (x,y) =& P \{ g(X) \le x, g(Y) \le y \} = P \{ X \in g^{-1} (-\infty,x], Y \in h^{-1}(-\infty, y] \} \\
    =& P \{X \in g^{-1} (-\infty,x] \} P \{Y \in h^{-1}(-\infty, y] \} = P \{g(X) \le x \} P \{h(Y) \le y \} \\
    =& F_{g(X)}(x) F_{g(Y)}(y)
\end{align*}
This concludes the proof.
Regards,
D