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 1,2,3 with P( 1 ) P( 2 ) P( 3 ) 1 3 Let G be the collection of all subsets of G , 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , Let x and y be random variables, and set ai x(i)for i 1,2,3 Assume no two of the ai are the same Suppose y(1) b1 and y(2) y(3) 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 field generated by y

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Question: 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

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˜ ?

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