Uncorrelated but utterly dependent

Question

Let X take the values -1, 0, 1 each with probability 	frac13, and define Y = X^2. What is the correlation between X and Y? Are they independent?

Accepted Answer

Correlation. First E[X] = 	frac{-1 + 0 + 1}{3} = 0. The covariance is \mathrm{Cov}(X, Y) = E[XY] - E[X]E[Y] = E[X \cdot X^2] - 0 = E[X^3]. Since X^3 takes values -1, 0, 1 with equal probability, E[X^3] = 0. So \mathrm{Cov}(X, Y) = 0 and the correlation is 0: X and Y are uncorrelated. Independence? Absolutely not, Y is a deterministic function of X. Knowing X pins down Y exactly: P(Y = 0 \mid X = 0) = 1 
e P(Y = 0) = 	frac13. You cannot get more dependent than "one variable determines the other," yet the correlation is exactly zero. Why this happens. Correlation measures only the linear rel…

Uncorrelated but utterly dependent

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