1. Problem (20 pts) (a) (2 pts) A biased coin is tossed, and it is assumed the chance of getting a head, H, is = (Thus the chance of getting a tail, T, is ). Consider a random experiment of throwing the coin FIVE times. Let S denote the sample space. Describe the elements of S. (b) ( 2 pts) Let X be the random variable that corresponds to the number of the heads coming up in 5 times of tossing. What are the values that the random variable X takes? (c) (2 pts) Find the probability that there is 3 tails and 2 heads, that is, P(X = 2). (d) (2 pts) Find the probability that there are at least 2 heads, that is, P (X 2 2) (e) ( 2 pts) Suppose that for each toss that coms up heads, we win $4, but for each toss that comes up tails, we lose $3. Clearly, the quantity of interest in this situation is our total winning. Let Y denote this quantity. What re the values the random variable Y can take? (f) ( 2 pts) Find P( Y = 6). (g) ( 2 pts) Find P (Y S 0) (h) (3 pts) Compute the expectation and the variance of the random variable X, that means E(X) = ? Var (X) =? (k) (3 pts) Compute the E(Y), Var(Y)