TITLE: Is there a *simple* example showing that uncorrelated random variables need not be independent?
QUESTION [30 upvotes]: 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?

REPLY [2 votes]: Another easy to explain on the run:
for $\theta$ Uniform [0 , 2$\pi$] set X = sin($\theta$), Y = cos($\theta$).
Then $X^2 + Y^2 = 1$ - clearly X and Y are not independent.
However as correlation is a linear relationship it is unable to produce anything but zero for the correlation between X and Y