A Universe of Sorts

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§ Example of Covariance Zero, and yet "correlated"

created 2021-10-31

$x$ 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 $corr(X, Y) = sgn(a)$ .

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