A point on a circle, zero correlation, total dependence

Question

Pick an angle 	heta uniformly on [0, 2\pi) and let X = \cos	heta, Y = \sin	heta, the coordinates of a uniformly random point on the unit circle. What is the correlation between X and Y? Are they independent?

Accepted Answer

Correlation. Over a full period, E[\cos	heta] = 0 and E[\sin	heta] = 0. For the covariance, \mathrm{Cov}(X, Y) = E[\cos	heta \sin	heta] = E\!\left[	frac12 \sin 2	hetaight] = 0, since \sin 2	heta also averages to zero over a full period. So X and Y are uncorrelated. Independence? No. The point lies on the unit circle, so X and Y obey the hard constraint X^2 + Y^2 = 1. Knowing X forces Y = \pm\sqrt{1 - X^2}, only two possible values instead of a whole range. For instance, X = 1 forces Y = 0 with certainty, whereas unconditionally Y is spread over [-1, 1]. That is about as dependent as …

A point on a circle, zero correlation, total dependence

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