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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?

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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.

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