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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...

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Basic Math

1. Can two variables share a deterministic relationship, andyet be uncorrelated? For instance, suppose that X follows acontinuous, 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 symmetricaround zero—it is clear that E(X) = 0.
Now let Y = X2.
b. Clearly Y and X are not independent; in fact, adeterministic relationship exists between them (if you know X, youknow 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 anobvious relationship?

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