§ Example of covariance zero, and yet "correlated"

• $x$ and $y$ coordinates of points on a disk.
• $E[X], E[Y]$ is zero because symmetric about origin.
• $E[XY] = 0$ because of symmetry along quadrants.
• Thus, $E[XY] - E[X] E[Y]$, the covariance, is zero.
• However, they are clearly correlated. Eg. if $x = 1$, then $y$ must be zero.
• If $Y = aX+b$ the $corr(X, Y) = sgn(a)$.