# Independence of squared random variables

Let $$X\sim M$$ and $$Y\sim N$$ be independent with $$M,N$$ being probability distributions. Does it always hold that for example $$X^2$$ and $$Y$$ (and analogous cases) are independent as well?

Yes. If $$f,g$$ are measurable maps and $$X,Y$$ are independent, then also $$f(X)$$ and $$g(Y)$$ are independent.
Proof: $$P(f(X) \in A, g(Y) \in B) = P(X \in f^{-1}(A), Y \in g^{-1}(B))$$ $$= P(X \in f^{-1}(A)) P(Y \in g^{-1}(B)) = P(f(X) \in A) P(g(Y) \in B)$$
where $$A, B$$ are Borel sets. $$\quad \square$$
Next, apply this with $$f: x \mapsto x^2, g: x \mapsto x$$ which are continuous maps, hence measurable.