If X and Y are iid random variables , $X^2$ and $Y^2$ are independent? If X and Y are iid  random variables, $X^2$ and $Y^2$ are independent?
In some problems I see that this is true, but is true in general?
 A: They don't need to be identically distributed : in general if $X$ and $Y$ are independent, and $f,g$ are Borel functions, then $f(X)$ and $g(Y)$ are independent. In particular if $f(x) = g(x) = x^2$ then $X^2$ and $Y^2$ would be independent.
In more generality, read up the disjoint blocks theorem , which is Theorem 3.46 here, which would assert something similar for an independent family of random variables and not just two. See example 3.49 for an example with more than just two random variables.
Of course, if $X,Y$ are in addition identically distributed, then $X^2$ and $Y^2$ will be iid.
A: Yes it's true ! In general, if $X_1,...,X_n$ and $Y_1,...,Y_n$ are independent and $f=\mathbb R^n\to \mathbb R$ and $g:\mathbb R^m\to \mathbb R$ are Borel functions, then $f(X_1,...,X_n)$ and $g(Y_1,...,Y_m)$ are independents.
The proof goes as follow : $\sigma (X_1,...,X_n)$ and $\sigma (Y_1,...,Y_m)$ are independents, and since $f(X_1,...,X_n)\in \sigma (X_1,...,X_n)$ and $g(Y_1,...,Y_m)\in\sigma (Y_1,...,Y_m)$, the claim follow.
