probability and random process

EXERCISE 2.1. Let ? = {a, b, c, d} and let F = 2" (the set of all subsets of $2). We define a probability measure P as follows P(a) = 1/6, P(b) = 1/3, P(c) = 1/4, P(d) = 1/4, Next, define three random variables X(a) =1, X(b) = 1, X(c) = -1, X(d) =-1, Y(a) = 1, Y(b) =-1, Y(c) = 1, Y(d) = -1, and Z = X + Y. (a) List the sets in o(X). (b) What are the values of E[Y|X] for {a, b, c, d}? Verify the partial averaging property: E[1, E[Y|X]] = E[1AY] for all A E o(X). (c) What are the values of E[Z X] for {a, b, c, d}? Verify the partial averaging property. probability and random process