Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$ Does anyone have an example of two dependent random variables, that satisfy this relation?
$E[f(X)f(Y)]=E[f(X)]E[f(Y)]$
for every function $f(t)$.
Thanks.
*edit: I still couldn't find an example. I think one should be of two identically distributed variables, since all the "moments" need to be independent: $Ex^iy^i=Ex^iEy^i$. That's plum hard...

If you take dependent random variables $X$ and $Y$, and set $X^{'} = X - E[X]$ and $Y^{'} = Y - E[Y]$, then $E[f(X^{'})f(Y^{'})]=E[f(X^{'})]E[f(Y^{'})]=0$ as long as $f$ preserves the zero expected value. I guess you cannot show this for all $f$.
