Consider the example of Liam and Martin's game. Namely, if {X, n>0} describes Liam's wealth at...

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Consider the example of Liam and Martin's game. Namely, if {X, n>0} describes Liam's wealth at step n, then Xn+1 = X + Jn+1, where the Jr's are independent and identically distributed. The distribution of Jn will change in the parts below; you must answer the questions based on the assumptions provided. = 2. [25 points] In the same setting of Liam and Martin's example; now assume that P(Jn 1) = p, P(Jn = 1) = q, P(Jn = 0) = r, with p+q+r = 1. Liam and Martin's total wealth is $7. The cost for playing the game is given by r(x) = 1 100 min(x, 7-x), that is 1 cent per dollar of the lower wealth of the players. This cost is paid separately, that is, the wealth of Martin and Liam is not used to pay for the cost. Define A = {0, 7}. a.) [5 points] Provide the transition probability matrix of {Xn: n>0}. b.) [5 points] Let C be the total cost of playing the game. That is C = r(Xn), n=0 where TA is the duration of the game. What is u(0) and u(7) if u(i) = E[C|X = i]. c.) [10 points] Provide a system of linear equation that can be used to solve for u(1), u(2), ..., u(6). In particular, find a matrix B of size 6-by-6 and a vector b such that if u = (u(1),..., u(6)), then Bu = b. Note: write B in terms of p, q, r. d.) [5 points] Suppose that Martin's initial wealth is $4; p = ,q = r = . What is the cost of playing the game? Consider the example of Liam and Martin's game. Namely, if {X, n>0} describes Liam's wealth at step n, then Xn+1 = X + Jn+1, where the Jr's are independent and identically distributed. The distribution of Jn will change in the parts below; you must answer the questions based on the assumptions provided. = 2. [25 points] In the same setting of Liam and Martin's example; now assume that P(Jn 1) = p, P(Jn = 1) = q, P(Jn = 0) = r, with p+q+r = 1. Liam and Martin's total wealth is $7. The cost for playing the game is given by r(x) = 1 100 min(x, 7-x), that is 1 cent per dollar of the lower wealth of the players. This cost is paid separately, that is, the wealth of Martin and Liam is not used to pay for the cost. Define A = {0, 7}. a.) [5 points] Provide the transition probability matrix of {Xn: n>0}. b.) [5 points] Let C be the total cost of playing the game. That is C = r(Xn), n=0 where TA is the duration of the game. What is u(0) and u(7) if u(i) = E[C|X = i]. c.) [10 points] Provide a system of linear equation that can be used to solve for u(1), u(2), ..., u(6). In particular, find a matrix B of size 6-by-6 and a vector b such that if u = (u(1),..., u(6)), then Bu = b. Note: write B in terms of p, q, r. d.) [5 points] Suppose that Martin's initial wealth is $4; p = ,q = r = . What is the cost of playing the game?