Question 4. Let (12,F,P) be a given probability space where N = {1, ... ,12}, F is the power set of 12, and P is the pro

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Question 4 Let 12 F P Be A Given Probability Space Where N 1 12 F Is The Power Set Of 12 And P Is The Pro 1 (118.62 KiB) Viewed 64 times

Question 4. Let (12,F,P) be a given probability space where N = {1, ... ,12}, F is the power set of 12, and P is the probability measure defined by P({w}) = 1 for all w. Define a discrete random variable X 1 2 X(w) = 3 if w=1,2; if w = 3,4; if w = 5,6; if w = 7,8; if w = 9, 10, 11, 12; 4 5 and let us consider a o-algebra G = {0, 12, A, B, C, AU B, AUC, BUC} where A = {1,2,3,4}, B = {5,6,7,8}, C = = {9, 10, 11, 12} (a) Is X G-measurable or not? (Prove your claim.) (b) Is a function Z:N + R defined by 1.5 Z(w) = = 3 if w= 1, 2, 3, 4; if w = 5, 6, 7, 8; if w = 9, 10, 11, 12 5 a conditional expectation of X given G or not? (Prove your claim.)