If $X$, $Y$ are independent, then $X^2$ and $Y^2$ are independent

Question

Is it true that if $X$, $Y$ are independent, then $X^2$ and $Y^2$ are independent?

I know that the first is true if and only if $P(X=k, Y=l) = P(X=k)P(Y=l)$ but I don't know anything about $X^2$ and $Y^2$

Answer 1

Note that (for $k,l>0$ - the other cases are similar) $$P(X^2=k,Y^2=l)=P(X=\sqrt k,Y=\sqrt l)+P(X=-\sqrt k,Y=\sqrt l)+P(X=\sqrt k,Y=-\sqrt l)+P(X=-\sqrt k,Y=-\sqrt l) $$ whereas $$P(X^2=k)P(Y^2=l)=(P(X=\sqrt k)+P(X=-\sqrt k))(P(Y=\sqrt l)+P(Y=-\sqrt l)) $$ If you expand this product, you get four summands of type $P(X=\pm\sqrt k)P(Y=\pm\sqrt l)$, which - by independece of $X,Y$ - match the summands in the first equation.

Answer 2

Let X, Y are independent with the values {-1, 1}...