If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$? [duplicate]

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