Are functions of independent variables also independent?

It’s a really simple question. However I didn’t see it in books and I tried to find the answer on the web but failed.

If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$ = $f_1(X_1)$ and $Y_2$ = $f_2(X_2)$.

Intuitively, $Y_1$ and $Y_2$ should be independent, and I can’t find a counter example, but I am not sure. Could anyone tell me whether they are independent? Does it depend on some properties of $f_1$ and $f_2$?

Thank you.

Answer

Yes, they are independent.

If you are studying rigorous probability course with sigma-algebras then you may prove it by noticing that the sigma-algebra generated by $f_{1}(X_{1})$ is smaller than the sigma-algebra generated by $X_{1}$, where $f_{1}$ is borel-measurable function.

If you are studying an introductory course – then just remark that this theorem is consistent with our intuition: if $X_{1}$ does not contain info about $X_{2}$ then $f_{1}(X_{1})$ does not contain info about $f_{2}(X_{2})$.

Attribution
Source : Link , Question Author : LLS , Answer Author : Roah

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