A die roll and its squared distance from the mean

Question

Roll a fair six-sided die and let X be the result. Define Y = (X - 3.5)^2, the squared distance of the roll from the average value 3.5. What is the correlation between X and Y?

Accepted Answer

Y is built directly from X, so they are dependent. Still, the correlation is 0, and symmetry is the reason. It helps to center the die: let Z = X - 3.5, which is symmetric around 0, taking the values -2.5, -1.5, -0.5, 0.5, 1.5, 2.5 each with chance 	frac16. Then Y = Z^2, and shifting X by a constant does not change any covariance, so \mathrm{Cov}(X, Y) = \mathrm{Cov}(Z, Z^2) = E[Z^3] - E[Z]\,E[Z^2]. The symmetric averages. Because Z is symmetric around zero, its average is 0, and the average of its cube is 0 too (each positive value cancels its negative twin): E[Z] = 0, \qquad E[Z^3] = 	fra…

A die roll and its squared distance from the mean

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