Exercise 5. Alice and Bob alternate playing at the casino table. (Alice starts and plays at odd times i = 1,3. ...; Bob plays at even times i = 2, 4. ...) At each time i, the net gain of whoever is playing is a random variable G; with the following PMF: 1/3, 9= -2, pc(g) = 1/2, g = 1, 1/6, 9 = 3. Assume that the net gains at different times are independent. We refer to an outcome of -2 as a "loss", and an outcome of 1 or 3 as a "win." (1) They keep gambling until the first time where a loss by Bob immediately follows a loss by Alice. Write down the PMF of the total number of rounds played. (A round consists of two plays, one by Alice and then one by Bob.) (2) Write down an expression for the transform of the net gain of Alice up to the first time of the first loss by Bob. (3) Write down the PMF for Z, defined at the time at which Bob has his third loss. (4) Let N be the number of rounds until each of them has won at least once. Find E[N]