Quick question - if $A$ and $B$ are independent continuous random variables, does that imply $A$ and $\cos B$ are also independent?
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1Yes, and continuity of $A$ and $B$ is not needed. – Nov 18 '15 at 17:39
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1is there a way to formally prove this? – Arron Nov 18 '15 at 17:40
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You only need to use this lemma in the case $f=id_\mathbb R,\ g=\cos(\bullet)$.
If $A$, $B$ are independent random variables and $f$, $g$ are borelian functions, then $f(A)$ and $f(B)$ are independent.
Indeed, let $E, F$ borel sets. Then, $$P(f(A)\in E\,\wedge\, g(B)\in F)=P(A\in f^{-1}(E)\,\wedge\, B\in g^{-1}(F))=\\=P(A\in f^{-1}(E))\cdot P(B\in g^{-1}(F))=P(f(A)\in E)\cdot P(g(B)\in F)$$