Verify the following properties of the conditional expectation. When in doubt, you are welcome to consider...

  

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Verify the following properties of the conditional expectation. When in doubt, you are welcome to consider only the particular case when the random variables X, Y have a joint density or pmf. (1) E(E(XY)) = EX; (2) E(g(Y)XY) = g(Y)E(XY) for every bounded measurable function g; (3) Cov(E(XY),Y) = Cov(X, Y); (4) Y and X-E(XY) are uncorrelated; (5) E(XY) = EX if X and Y are independent. Verify the following properties of the conditional expectation. When in doubt, you are welcome to consider only the particular case when the random variables X, Y have a joint density or pmf. (1) E(E(XY)) = EX; (2) E(g(Y)XY) = g(Y)E(XY) for every bounded measurable function g; (3) Cov(E(XY),Y) = Cov(X, Y); (4) Y and X-E(XY) are uncorrelated; (5) E(XY) = EX if X and Y are independent.