If $X$ and $Y$ are independent, how to proof $X^2$ and $Y^2$ are also independent

Question

this is Probability Density Function(pdf) if $X$ and $Y$ are independent, how to prove $X^2$ and $Y^2$ are also independent

Answer 1

That $X$ and $Y$ are independent just means that the sigma algebras they generate are independent. But the sigma algebra $X^2$ generates is containedin the sigma algebra generates by $X$ and the same goes for $Y^2$.

Answer 2

Here is another way, if you are not familiar with measure theory. I assume we are talking about real-valued random variables. Let $a,b\geq0$ $$P(X^2 \leq a,\,Y^2 \leq b)=P(-\sqrt{a}\leq X \leq \sqrt{a},\,-\sqrt{b}\leq Y \leq \sqrt{b}) \\= P(-\sqrt{a}\leq X \leq \sqrt{a})P(-\sqrt{b}\leq Y \leq \sqrt{b})=P(X^2 \leq a)P(Y^2 \leq b),$$ where we have used the independence of $X,Y$ at the beginning of the second line. If $a<0$ or $b<0$, then $$P(X^2 \leq a,\,Y^2 \leq b)=0=P(X^2 \leq a)P(Y^2 \leq b).$$ Hence, $X^2,Y^2$ are independent.