Question: Prove that if E[Y|X = x] = E[Y] for all x, then X and Y are uncorrelated; give a counterexample to show that the converse

Prove that if E[Y|X = x] = E[Y] for all x, then X and Y are uncorrelated; give a counterexample to show that the converse is not true.
Prove and use the fact that E[XY] = E[XE[Y|X]].

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