1. (35 points) Suppose we are interested in analyzing the market share and customer loyalty for Murphy's Foodliner (1), Ashley's Supermarket (2), and Quick Stop Groceries (3), the three grocery stores in a small town. We focus on the sequence of shopping trips of one customer and assume that the customer makes one shopping trip each week to either Murphy's Foodliner, Ashley's Supermarket, or Quick Stop Groceries, but not all. Quick Stop Groceries is smaller than either Murphy's Foodliner or Ashley's Supermarket. However, Quick Stop's convenience with faster service and gasoline for automobiles can be expected to attract some customers who currently make weekly shopping visits to either Murphy's or Ashley's. Assume that the transition probabilities are as follows: Current Purchase 0.85 0.20 0.15 Next Purchase 0.10 0.75 0.10 0.05 0.05 0.75 a) (10 points) Define X, and write down the state space S. b) (5 points) Draw the transition diagram. c) (10 points) Could we model the market share and customer loyalty of three grocery stores as a Markov Chain? Explain. d) (10 points) Do you think the stationarity assumption is reasonable for your model? Explain why or why not. Also, indicate with an arrow where you would put the stationarity assumption in the following scale in this real-life situation: Totally False Totally Correct