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...

The relation $E[XY]=E[X]E[Y]$ holds if and only if the covariance of $X$ and $Y$ is zero, that is, $X$ and $Y$ are uncorrelated.
So, you are actually asking is there dependent but uncorrelated random variables $X$ and $Y$ such that every function $f$ would preserve the uncorrelatedness. 
Consider a function $f(x) = a$, where $a$ is some constant. Now, $f(X) = a$ and $f(Y) = a$, and therefore $f(X)$ and $f(Y)$ are correlated. Thus, the answer to your question is that there are no such variables.
