Question: 3. (40 points total) Consider a random vector X which is made up of two random vari- ables (X1, X) where each random variable X;

3. (40 points total) Consider a random vector X which is made up of two random vari- ables (X1, X) where each random variable X; has two possible outcomes: 1 and +1. (a) What is the sample space 2 of possible outcomes/events and the sigma-algebra generated by X, F = o(X)? (b) How can we characterize measures p, probability measures P, and measurable functions on this measurable space (2, F)? (c) Given a probability measure P, how can we construct another absolutely continuous probability measure ()7 (d) Given a probability measure P, suppose we define a new random variable Z = X, + Xso. What is the distribution function F of Z7? (e) Is Z integrable? If so, what is its expectation F(Z)

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