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Show all of your work, and upload this homework to Canvas. 1. Every day, Eric takes the same street from his home to the university. There are 4 street lights along his way, and Eric has noticed the... following Markov dependence. If he sees a green light at an intersection, then 60% of time the next light is also green, and 40% of time the next light is red. However, if he sees a red light, then 70% of time the next light is also red, and 30% of time the next light is green. Let 1 = “green light” and 2 = “red light” with the state space {1, 2}. (a) Construct the 1-step transition probability matrix for the street lights. (b) If the first light is green, what is the probability that the third light is red? (c) Eric’s classmate Jacob has many street lights between his home and the university. If the first street light is green, what is the probability that the last street light is red? (Use the steady-state distribution.) Answer: Stat 330 (Spring 2015): Homework 8 Due: March 27, 2015 4. (Baron’s book): 6.3 Answer: 5. * (Baron’s book): 6.4 Answer: 6. (Baron’s book): 6.7 Answer: 3 2. We want to model the daily movement of a particular stock (say Amazon, ticker = AMZN) using a homogenous markov-chain. Suppose at the close of the market each day, the stock can end up higher or lower than the previous day’s close. Assume that if the stock closes higher on a day, the probability that it closes higher the next day is .65. If the stock closes lower on a day, the probability that it closes higher the next day is .45. (a) What is the 1-step transition matrix? (Let 1 = higher, 2 = lower) Answer: P =  1 2 1 0.65 0.35 2 0.45 0.55  (b) On Monday, the stock closed higher. What is the probability that it will close higher on Thursday (three days later) Answer: P (3) = P · P · P =  1 2 1 0.566 0.434 2 0.558 0.442  The distribution of X three steps ahead is: P3 = P0 · P (3) = (1, 0) ·  0.566 0.434 0.558 0.442 = (0.566, 0.434) So, the probability it closes higher Thursday (three steps in the future) is 0.566. [Show More]

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