law of iterated expectations for continuous random variable

Question

 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.