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

1. Can two variables share a deterministic relationship, and yet be uncorrelated? 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 a symmetry argument—the distribution of X is symmetric around zero—it is clear that E(X) = 0.

Now let Y = X2.

b. Clearly Y and X are not independent; in fact, a deterministic relationship exists between them (if you know X, you know Y). Show, however, that X and Y are uncorrelated: Cov(X,Y) = 0. Hint: use a symmetry argument for the distribution of X3.

c. Can you explain why there is no correlation despite an obvious relationship?

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orchestra answered 1 year ago

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