Passport is a consumer credit card company that has a large number of cus- tomers (or accounts). These customers charge some of their purchases on their Passport cards. The charges made in one month are due by the end of the next month. If a customer fails to make the minimum payment in a given month, the company flags the account as delinquent. The company keeps track of the payment history of each customer so that it can identify customers who are likely to default on their obligations and not pay their debt to the company. Here, we describe the simplest method by which a passport tracks its accounts. A customer is said to be in state (or delinquency stage) k if he or she has missed making the minimum payment for the last k consecutive months. A customer in state k has four possible futures: make a minimum payment (or more) and move to stage 0, make no payment (or less than the minimum payment) and move to stage k+1, default by declaring bankruptcy, and after some legal procedures move to stage 0, or the company can cancel the customer's card, terminate the account, and find a new customer. In this case, we can assume that the customer also moves to stage 0. Currently, the company has a simple policy: it terminates an account as soon as it misses seven minimum payments in a row and writes off the remaining outstanding balance on that account as a loss. That is, for an account in state k < 6 the company can lose the outstanding balance with a certain probability. From the experience with the customers over the years, the company has esti- mated: the probability that a customer in state k fails to make the minimum payment in the current period and thus moves to state k + 1, or the account is terminated if k = 6. the probability that a customer in state k declares bankruptcy in the current period and thus moves to state 0. the average outstanding balance of a customer in state k. These parameters for 0 k 6 are given in Table 1. k 0 1 2 3 4 5 6 P(fails to pay in state k) 0.073 0.062 P(bankruptcy in state k) 0.058 0.022 0.054 0.042 0.131 0.02 0.184 0.283 0.326 0.041 0.139 0.14 Average balance in state k 1,682.25 2,250.87 2.330.63 2,674.12 3.213.45 3,492.39 4.218.35 Table 1: Data for Passport account holders. Task 1. Let X be the state of a customer at time n. Justify that {Xn, n > 0} is a Markov chain. Using the data from Table 1, construct the 1-step transition probability matrix.