Is there a simple example showing that given $X,Y$ uncorrelated (covariance is zero), $X,Y$ are not independent?
I have looked up two references, however, I am dissatisfied with both.
In Reference $1$, $X,Y$ are assumed to be independent uniform RVs from $(0,1)$, construct $Z = X+Y, W = X - Y$, then the claim is that $Z,W$ is uncorrelated but not independent. Unfortunately, finding the PDF of $Z,W$ is not trivial.
In Reference $2$, $\phi$ is assumed to be uniform RV from $(0, 2\pi)$, and construct $X = \cos(\phi)$, $Y = \sin(\phi)$. Then the claim is that $X,Y$ are uncorrelated but not independent. Unfortunately, the PDFs of $X,Y$ takes on the form of rarely mentioned arcsine distribution.
I just wish to have an example at hand where I can whip out to show that uncorrelated does not necessarily implies independent. Is this do-able?