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Page of414 O 200M + SECTION 10.3 PROBLEM SET: REGULAR MARKOV CHAINS 1) Determine whether the following matrices are regular Markov chains. all .3] ml?)

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Page of414 O 200M + SECTION 10.3 PROBLEM SET: REGULAR MARKOV CHAINS 1) Determine whether the following matrices are regular Markov chains. all .3] ml?) '1'] .6 o .4 .2 .4 .4 c) .2 .4 .4 d) .6 .4 o o o o .3 .2 .5 2) Company I and Company II compete against each other. and the transition matrix for people switching from Company I to Company 11 is given below. To Company I Company II FROM Company I .3 .7 Company II .8 .2 a) If the initial market share is 40% for CompanyI b) If this trend continues, what is the long range and 60% for Company 11. what will the market expectation for the market? share be after 3 transitions? 3) Suppose the transition matrix for the tennis player in Exercise 4 of the last section is as follows, Where C denotes the crossrourt shots and D denotes downthe-line shots. Next Shot C D Previous C .9 .1 Shut D .7 .3 3) Suppose the transition matrix for the tennis player in Exercise 4 of the last section is as follows, where C denotes the crosscourt shots and D denotes downthe-line shots. Next Shot C D Previous C '9 .1 Shot D '7 .3 a) If the player hit the rst shot cross-court. what b) Determine the long term shot distribution. is the probability he will hit the fourth shot cross-court'.1 367 5) In a bikeshare program with 3 bike stations. A. B. and C, people can borrow a bicycle atone station and return it to the same station or either of the other two stations. The transition matrix is: A B C T A 0.3 05 02 B 0.1 06 03 C 0.1 0.1 0.8 Find the following: a) If a bicycle is initially at station A, what is the b) If the initial distribution of bicycles is 50% at probability it will be at station C aer 5 days? station A. 20% at station B. and 30% at station C. what will be the distribution after 2 days? Aer 5 days? c) What will be the eventual long term distribution d) If initially the distribution of bicycles at the of bicycles at the stations? stations was evenly distributed With one third of the bicycles at each station, will the eventual long term distribution be different than if the initial distribution is as given in part (c)? any Page of414 SECTION 10.3 PROBLEM SET: REGULAR MARKOV CHAINS 6) Suppose that a country has 3 political parties: the Conservative (C), Liberal (L), and National (N) parties. If a person votes for the candidate 'om one party in an election, that person may decide to vote for the same party in the next election or may switch to vote for a candidate om another party in the next election. The transition matrix is: NEXT ELECTION C L N C .5 .4 .1 THIS L .3 .4 .3 ELECTION N .2 .2 .6 Assume there is an election every year, so the transition time is one year. Find the following. a) If a person voted for the Liberal party in this b) If a person voted for the National party in this election, nd the probability that the person election, nd the probability that the person votes for the National party in the next election. votes for the Conservative party in the election two years from now. c) If in this election Conservatives received 25% d) Assuming the current distribution from part (c), of the votes, Liberals 30% of the votes, and what will the distribution be in the election two Nationals the remaining 45% of the votes, years from now? What is the predicted distribution for the next election? O ZOOM + Page of414 Assume there is an election every year, so the transition time is one year. Find the following. a) If a person voted for the Liberal party in this b) If a person voted for the National party in this election, nd the probability that the person election, nd the probability that the person votes for the National party in the next election. votes for the Conservative party in the election two years from now. c) If in this election Conservatives received 25% d) Assuming the current distribution from part (c), of the votes, Liberals 30% of the votes, and what will the distribution be in the election two Nationals the remaining 45% of the votes, years from now? what is the predicted distribution for the next election? e) Assuming the current distribution from part (c), 1) Determine the long term distribution. what will the distribution be in the election three years from now? O ZOOM +

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