Let X and Y be two random variables and
$E[f_1(X), f_2(Y)] = E[f_1(X)] *E[f_2(Y)]$
then can I conclude that $f_1(X)$ and $f_2(Y)$ are independent?
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$f_1(X)$ and $f_2(Y)$ are always independent given that $X$ and $Y$ are independent. – Nov 15 '19 at 14:21
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@d.k.o. edited, sorry – hanugm Nov 15 '19 at 14:24
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Take $X=Z$ and $Y=Z^2$, where $Z\sim N(0,1)$. Then $\mathsf{E}XY=\mathsf{E}Z^3=0=\mathsf{E}X\mathsf{E}Y$. In your case $f_1(X)$ and $f_2(Y)$ are just uncorrelated.
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If your equation $E[f(X)g(Y)]=(E[f(X)])(E[g(Y)])$ holds for enough different functions the answer becomes yes. Such as, the set of all $f,g$ of form $x\mapsto\mathbb 1_{[a,b]}(x)$, or the set of all continuous bounded $f,g$, or the set of all functions of form $u\mapsto\exp(au)$, etc.
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