1. A businessman travels between the cities of the following graph. The travel pattern is that...

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1. A businessman travels between the cities of the following graph. The travel pattern is that he stays in one of the cities for the whole day, and on the next day, he travels to one of the neighboring cities with equal probabilities. That is: if d is the degree of the vertex (city), then he travels to each of the neighboring cities with probability 1. At time/day 0, the businessman is positioned at city 1. (a) Find the expected number of jumps it takes the businessman to reach state 4. (b) Whenever the businessman visits city i, he makes a daily profit of R(i) := 5 - i dollars. Find his long-term average profit, namely: lim // 1 R(Xk). n k=1 (c) for this part, assume that after one year, the government builds a new road between city 2 and city 3 (add one more bridge between 2 and 3) Assume that the distance between cities i and j is - |ji|4 miles. Find the expected distance the businessman travels, until the businessman reaches state 4 for the first time. FOR M ISSI 1 3 4 (7+7+7 21 points) 1. A businessman travels between the cities of the following graph. The travel pattern is that he stays in one of the cities for the whole day, and on the next day, he travels to one of the neighboring cities with equal probabilities. That is: if d is the degree of the vertex (city), then he travels to each of the neighboring cities with probability 1. At time/day 0, the businessman is positioned at city 1. (a) Find the expected number of jumps it takes the businessman to reach state 4. (b) Whenever the businessman visits city i, he makes a daily profit of R(i) := 5 - i dollars. Find his long-term average profit, namely: lim // 1 R(Xk). n k=1 (c) for this part, assume that after one year, the government builds a new road between city 2 and city 3 (add one more bridge between 2 and 3) Assume that the distance between cities i and j is - |ji|4 miles. Find the expected distance the businessman travels, until the businessman reaches state 4 for the first time. FOR M ISSI 1 3 4 (7+7+7 21 points)