1. Independence of random variables can be affected by changes of measure. To illustrate this point, consider the space of two coin tosses 02 = {HH, HT, TH, TT} and let the stock price be given by So = 4, S,(H) = 8, S, (T) = 2 S2(HH) = 16, S,(HT) = S2(TH) = 4, S2(TT) = 1. Consider two probability measures given by P(TH) = P(TT) = 4' P(HH) = , P(HT) = P(HH) = P(HT) = . P(TH) = ,P(TT) = Define the random variable if So = 4 X = o if S2 # 4. a. List all the sets in o(X). b. List all the sets in o ($1). c. Show that (X) and o($1) are independent under the probability measure P. d. Show that o(X) and o(S1) are NOT independent under the proba- bility measure P e. Under P, we have P[S1 = 8] = ; and P[S1 = 2] = . Explain intuitively why, if you are told that X = 1, you would want to revise your estimate of the distribution of S1