Coordinates of a random compass point with zero correlation

Question

A dot lands on one of four points, each with chance 	frac14: East (1, 0), North (0, 1), West (-1, 0), and South (0, -1). Let X and Y be its two coordinates. What is the correlation between X and Y?

Accepted Answer

The coordinates are clearly linked: on any of these points, if X is nonzero then Y must be zero, and vice versa. Yet the correlation is 0. Show the covariance vanishes: \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y]. The two means. By the left-right and up-down symmetry of the four points: E[X] = 	frac14(1 + 0 - 1 + 0) = 0, \qquad E[Y] = 	frac14(0 + 1 + 0 - 1) = 0. The average of the product. At every one of the four points, at least one coordinate is zero, so the product XY is 0 at each point: E[XY] = 	frac14(1\cdot 0 + 0\cdot 1 + (-1)\cdot 0 + 0\cdot(-1)) = 0. Combine. \mathrm{Cov}(X, Y) = 0 - …

Coordinates of a random compass point with zero correlation

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