it is stochastic calculus and finance

We consider the measurable space (,F) where F=P(), corresponding to the experiment that consists of tossing a coin three consecutive times, each toss giving either "Head" (H) or "Tail" (T). We define the stock prices (Sn)0n3 on as follows: We define the probability measures P and P on (,F) by P()=81 for all , and P()=(53)k(52)3k where k is the number of " H " appearing in . We define the random variable X on as follows: X()={10ifS3()=4ifS3()=4 (2.1) Determine (X) and (S1) explicitly. (14) (2.2) Show that (X) and (S1) are independent under the probability measure P. (2.3) Show that (X) and (S1) are not independent under the probability measure P~. We consider the measurable space (,F) where F=P(), corresponding to the experiment that consists of tossing a coin three consecutive times, each toss giving either "Head" (H) or "Tail" (T). We define the stock prices (Sn)0n3 on as follows: We define the probability measures P and P on (,F) by P()=81 for all , and P()=(53)k(52)3k where k is the number of " H " appearing in . We define the random variable X on as follows: X()={10ifS3()=4ifS3()=4 (2.1) Determine (X) and (S1) explicitly. (14) (2.2) Show that (X) and (S1) are independent under the probability measure P. (2.3) Show that (X) and (S1) are not independent under the probability measure P~