Consider the following Markov chain model of an intemet retailer's marketing database. The state represents the classication of a customer at the beginning of a month. The set of possible states is {P, I, R, F, U} where: P represents a potential customer who has never purchased; I, R and F represent customers who purchase infrequently, regularly and frequently respectively; and U represents a customer who is considered unlikely ever to purchase again. The transition matrix modelling the change in the state of the process from the beginning of one month to the beginning of the next is: P I R F U P 0.6 0.2 0.2 0 0 P: I f 0 0.4 0.3 0 0.3\\ R 0 0.1 0.5 0.2 0.2 ' F K0 0 0.2 0.8 0] U 0 0 0 0 1 b) Calculate the probability that a potential customer in the database is ever classied as purchasing frequently in the future? c) Consider a customer in the database who is classied as purchasing frequently. Calculate the probability that such a customer is classied as purchasing frequently three months later. d) A new customer is added to the database and classied as a potential customer. Calculate the expected time until this customer is classied as unlikely to make further purchases. e) Does the equilibrium distribution for this Markov chain provide useful insight about the company's database? Explain your answer. f) Assume that customers who are classied as unlikely to make further purchases are immediately replaced, so that the total number of potential and active customers in the database remains constant. When replacement customers are added to the database, they are classied as potential customers. Model the evolution of the population of potential and active customers in the database as a Markov chain. Find the percentage of potential and active customers in each of the classications in the long run