4.1 (Tower rule for conditional expectations). Let (U, V1, V2, ..., Vn) denote a collection of jointly distributed random variables. The notation E[U|????1, ...,????n] denote the expected value of U conditional on (V1, ..., Vn)

taking the values (????1, ...,????n). The notation E[U|V1, ..., Vn] = W, say, denotes this conditional expectation, treated as a random variable depending on (V1, ..., Vn). Prove the following two important properties of W.

(a) E[W|V1, ..., Vn] = W, since W is constant given (V1, ..., Vn).

(b) E{E[W|V1, ..., Vn]} = E[W] = E[U], that is, the expectation of a conditional expectation is the same as the original expectation. This result for expectation is sometimes known as the Tower law.