Prove the law of iterated expectations for continuous random variables.
2. Prove that the bounds in Chebyshev's theorem cannot be improved upon. I.e., provide a distribution that satisfies the bounds exactly for k ≥1, show that it satisfies the bounds exactly, and draw its PDF. Then explain why, logically, this is the same as providing that the bounds cannot be improved upon.
3. In a logit model ln (p(X;Z) / (1-p(X;Z)) ) = α + β1X + β2Z, explain why the marginal effect of X on Y is a function of Z, even though no interaction term between Z and X is present.