A prepare the matrix of transition probabilities b

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Markov process with two states: city and suburbs. a. Prepare the matrix of transition probabilities. b. Compute the steady-state probabilities. c. In a particular metropolitan area, 40% of the population lives in the city, and 60% of the population lives in the suburbs. What population changes do your steady-state probabilities project for this metropolitan area? 8. Assume that a third grocery store, Quick Stop Groceries, enters the market share and customer loyalty situation described in Section 16.1. 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: a. Compute the steady-state probabilities for this three-state Markov process. b. What market share will Quick Stop obtain? 23610_ch16_ptg01_Web.indd 19 01/10/14 6:20 PM
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16-20 Chapter 16 Markov Processes c. With 1000 customers, the original two-state Markov process in Section 16.1 projected 667 weekly customer trips to Murphy’s Foodliner and 333 weekly customer trips to Ashley’s Supermarket. What impact will Quick Stop have on the customer visits at Murphy’s and Ashley’s? Explain.   9. The purchase patterns for two brands of toothpaste can be expressed as a Markov process with the following transition probabilities: To From Special B MDA T-White Special B 0.80 0.10 0.10 MDA 0.05 0.75 0.20 T-White 0.40 0.30 0.30 To From Special B MDA Special B 0.90 0.10 MDA 0.05 0.95 a. Which brand appears to have the most loyal customers? Explain. b. What are the projected market shares for the two brands? 10. Suppose that in Problem 9 a new toothpaste brand enters the market such that the follow- ing transition probabilities exist: What are the new long-run market shares? Which brand will suffer most from the intro- duction of the new brand of toothpaste? 11. In American football, touchdowns are worth 6 points. After scoring a touchdown, the scoring team may subsequently attempt to score one or two additional points. Going for one point is virtually an assured success, while going for two points is successful only with probability p . Consider the following game situation. The Temple Wildcats are los- ing by 14 points to the Killeen Tigers near the end of regulation time. The only way for Temple to win (or tie) this game is to score two touchdowns while not allowing Killeen to score again. The Temple coach must decide whether to attempt a 1-point or 2-point con- version after each touchdown. If the score is tied at the end of regulation time, the game goes into overtime where the fi rst team to score wins. The Temple coach believes that there is a 50% chance that Temple will win if the game goes into overtime. The probabil- ity of successfully converting a 1-point conversion is 1.0. The probability of successfully converting a 2-point conversion is p .
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  • Spring '18
  • Markov process, Markov chain, Andrey Markov, Markov decision process

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