I'm trying to conceptually understand questions 1 and 2. Basically it asks to prove that a conditional expectation has a lower variance than the unconditional random variable. (See screen shot)Let X_1, and X_2 be random variables that have a joint pdf fX1, fX2 (x1, x2). Suppose both have a finite variance.1. Prove that E[E[X_2|X_1]] = E[X_2]2. Now prove VE[X_2|X_1]]

In this challenge question, prove in the general case, that a conditional expectation has a lower variance than the unconditional random variable. That, is, let X1 and X2 be random variables that have the have a joint pdf JOY, X,($1, T2). Suppose that both variables have finite variance. (1) Prove that E[E[X2\\Xi]] = E[X2]. This probably feels familiar. (2) Using the statement that you proved in part 3.1, now prove that V[E[X2|Xi]]