## § 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 $corr(X, Y) = sgn(a)$.