The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so  =

{ω1,ω2,ω3} with P({ω1}) = P({ω2}) = P({ω3}) = 1/3. Let G be the collection of all subsets of :

G = {∅,{ω1},{ω2},{ω3},{ω1,ω2},{ω1,ω3},{ω2,ω3},}.

Let x˜ and y˜ be random variables, and set ai = ˜x(ωi)for i = 1,2,3. Assume no two of the ai are the same. Suppose y˜(ω1) = b1 and y˜(ω2) = ˜y(ω3) = b2 = b1.

(a) What is prob(x˜ = aj | ˜y = bi) for i = 1,2 and j = 1,2,3 ?

(b) What is E[˜x | ˜y = bi] for i = 1,2 ?

(c) What is the σ–field generated by y˜ ?