A number and its absolute value with zero correlation

Question

Let X be equally likely to be -1, 0, or 1. Define Y = X, the distance of X from zero. What is the correlation between X and Y?

Accepted Answer

Since Y is X with its sign stripped off, knowing X pins down Y exactly. Still, the correlation comes out to 0, because the link is not a straight line. Show the covariance vanishes: \mathrm{Cov}(X, Y) = E[XY] - E[X]\,E[Y]. The mean of X. By the symmetry of -1, 0, 1: E[X] = 	frac13(-1 + 0 + 1) = 0. So the covariance is just E[XY]. The average of the product. Here XY = X\,X: E[XX] = 	frac13\big((-1)(1) + (0)(0) + (1)(1)\big) = 	frac13(-1 + 0 + 1) = 0. The positive and negative sides cancel exactly. Combine. \mathrm{Cov}(X, Y) = 0, \qquad ho = 0. Yet the two are dependent: X = 0 forces Y = …

A number and its absolute value with zero correlation

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