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answerhappygod
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1. A traveler travels among four cities {1,2,3,4} every day. When the traveler is in city i, with probability 0 <p<1 (independent of everything else) she travels to city j the next day with probability proportional to the distance between i and j. With rest of the probability 1-p, she stays another day in the city. The distances between cities are given below: city 1 2 3 4 1 - 56 3 2 5 54 3 6 5 4 4 3 4 4 - a.) Construct a Markov chain model and list the detailed balance conditions. b.) Find the long run proportion of days that the travel visits each city. c.) Show how one can use the answer from part b.) to approximate the probability that the traveler is in city 3 after two thousand days, starting from city i. d.) Starting from city 1, what is the average time it takes for the traveler to revisit city 1 again? e.) Assume that city 4 has lost connection with the other 3 cities, and the traveler now only travels among cities {1,2,3). The transition probability of traveling to city 4 from any other city is transferred to the probability of staying in that city. In other words, the transition probabilities are defined as: P'li,j) = P(i,j) and i #j (1) P'(i,i) = P(1,4). Find the long run proportion of time only using your answer in (b).
1. A traveler travels among four cities {1,2,3,4} every day. When the traveler is in city i, with probability 0 <p<1 (independent of everything else) she travels to city j the next day with probability proportional to the distance between i and j. With rest of the probability 1-p, she stays another day in the city. The distances between cities are given below: city 1 2 3 4 1 - 56 3 2 5 54 3 6 5 4 4 3 4 4 - a.) Construct a Markov chain model and list the detailed balance conditions. b.) Find the long run proportion of days that the travel visits each city. c.) Show how one can use the answer from part b.) to approximate the probability that the traveler is in city 3 after two thousand days, starting from city i. d.) Starting from city 1, what is the average time it takes for the traveler to revisit city 1 again? e.) Assume that city 4 has lost connection with the other 3 cities, and the traveler now only travels among cities {1,2,3). The transition probability of traveling to city 4 from any other city is transferred to the probability of staying in that city. In other words, the transition probabilities are defined as: P'li,j) = P(i,j) and i #j (1) P'(i,i) = P(1,4). Find the long run proportion of time only using your answer in (b).