1. Alice and Bob alternate playing at the casino table. (Alice starts ...

1. Alice and Bob alternate playing at the casino table. (Alice startsand 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 arandom variable Gi with the following PMF:P G (g){1, g = −2312 , g = 116 , g = 30 , otherwiseAssume that the net gains at different times are independent. Werefer to an outcome of -2 as a “loss.”(a) They keep gambling until the first time where a loss by Bobimmediately follows a loss by Alice. Write down the PMF of thetotal number of rounds played. (A round consists of two plays,one by Alice and then one by Bob.)(b) Write down the PMF for Z, defined as the time at which Bob hashis third loss.(c) Let N be the number of rounds until each one of them has wonat least once. Find E(N).Solution(a)For each round, the probability that Alice and Bob both have a loss is 1 9 .Let random variable X represent the total number of rounds played untilthe first time where they both have a loss. Then X is a geometric randomvariable with parameter p = 1 9and has the following PMF.p X (x) = (1 − p) x−1 p = ( 8 x−19 ) ( 1 ) , x = 1,2, …9

(b)Consider the number of games M Bob played until his third loss.p M (m) = ( m − 1 33 − 1 ) (1 3 ) ( 2 m−33 ) , m = 3,4,5, …We define the other random variable Z as the time at which Bob has histhird loss. Z = 2M. By changing variables, we obtainzp Z (z) = ( 2 − 133 − 1 ) (1 3 ) ( 2 z3 ) 2 −3 , z = 6,8,10, …(c)Consider the following partitionA 1 : both win first roundA 2 : only Alice wins first roundA 3 : only Bob wins first roundA 4 : both lose first roundE[N] = E[N|A 1 ]P(A 1 ) + E[N|A 2 ]P(A 2 ) + E[N|A 3 ]P(A 3 )+ E[N|A 4 ]P(A 4 )E[N] = 1 ∙ ( 2 3 ∙ 2 3 ) + (1 + 1 ) ∙ ( 1 2 3 ∙ 2 3 ) ∙ 2 + (1 + E[N]) ∙ (1 3 ∙ 1 3 )3E[N] = 158