Question

1. Data from customer reward cards at a major grocery store chain show that of all customers who purchase Coke in their weekly grocery shopping, 90% will purchase it again the next week while 10% will purchase Pepsi instead. Of all customers who purchase Pepsi one week, 80% will purchase it again the next week and 20% will purchase Coke. Assuming that the probability that any given customer purchases Coke (or Pepsi) in any week depends only on their purchase in the previous week, model a customer's weekly Coke and Pepsi purchasing as a Markov chain and answer the following questions. b) Determine the one-step transition matrix, P, for the Markov chain. c) If a customer purchases Pepsi this week, what is the probability that she will purchase Coke two weeks from now? d) Write out and solve the equations for the steady-state probabilities: TT P = Tt and XEM TT= 1. e) Based on your answer to part d), what will Coke's market share be one year from now? f) Write out and solve the equations for the expected first passage times: Wij = 1 + Ekj Pik Ukj. g) For the customers who purchased Pepsi last week, what is the average number of weeks in a row that they will purchase Pepsi before switching to Coke? Explain your answer.

Answer 1

ANSWER :

b)

The one-step transition matrix is following

To? Coke Pepsi ? from Coke 0.3 Pepsi 0.2

c) Probability that the customer will purchase Coke next week = 0.2

Probability that the customer will purchase Coke two weeks from now = 0.2*0.9+0.8*0.2 = 0.34

d) Calculation of steady state probabilities is following :

Transition matrix T-1 To from Coke Pepsi Coke 0.9 0.2 Pepsi 0. 1 0.8 1 -1 0 -0.1 1 | = | 0.2 0.1 -0.2 0 RHS Coke -0.1 0.1 Pepsi 0.2 -0.2 S* (T-1) => 0 SO => 1 1 1 1 0.2(3) + (2) => 0.1(3) + (1) => 0.3 0.0 0.0 0.3 0.20 0.10 ---- ---- Steady State Probabilities (4) => 71 or P(Coke) = (5) => T2 or P(Pepsi) = 0.667 0.333

e) Coke's market share = 66.7 %

Pepsi's market share = 33.3%