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  3. 1. can two variables share a deterministic relatio

1. Can two variables share a deterministic relationship, and yet be uncorrelated? For instance, suppose that X...

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Statistics


1. Cantwo variables share a deterministic relationship, and yet beuncorrelated? For instance, suppose that X follows a continuous,uniform distribution between −1 and 1:
X ~U[−1, 1].
a.Draw a graph of the distribution of X.
By asymmetry argument—the distribution of X is symmetric around zero—itis clear that E(X) = 0.
Nowlet Y = X2.
b.Clearly Y and X are not independent; in fact, a deterministicrelationship exists between them (if you know X, you know Y). Show,however, that X and Y are uncorrelated: Cov(X,Y) = 0. Hint: use asymmetry argument for the distribution of X3.
c. Canyou explain why there is no correlation despite an obviousrelationship?

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