160a09_hw7sol - Pstat160 Homework#5 Solution Winter 2009...

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Pstat160 Winter 2009 Homework #5 Solution Problem 1. pb 24 Define the state to be the color of the ball just selected, and let 1=red, 2=white, 3=blue. To find the transition probability say P 11 , we need to find the probability to select a red ball from urn red, which is 1 4+1 = 1 / 5. We can find all transitions the same way and get P = 1 / 5 0 4 / 5 2 / 7 3 / 7 2 / 7 3 / 9 4 / 9 2 / 9 We need to find the limiting probabilities π . The set of linear equations are π (1) = 1 / 5 π (1) + 2 / 7 π (2) + 3 / 9 π (3) π (2) = 3 / 7 π (2) + 4 / 9 π (3) 1 = π (1) + π (2) + π (3) To solve these we use the second equation to find π (2) = 7 9 π (3) and plug this into first equation to get get π (1) = 25 36 π (3). Using the last equation, we finally get π (1) = 25 89 , π (2) = 28 89 π (3) = 36 89 This gives the respective proportion of selected colors in the long run. Problem 2. pb 30 We can think of this problem as fixing a position on the highway and recording the type of the last vehicle to pass. So X n will be a Markov chain with two states; Car (1) and truck (2) The probability to see a truck after a car is 1/5, probability of a car after a truck is 3/4. This gives the transition matrix P = 4 / 5 1 / 5 3 / 4 1 / 4 ! We need to find the limiting probabilities π (1) and π (2). The set of linear equations are π (1) = 4 / 5 π (1) + 3 / 4 π (2) π (2) = 1 / 5 π (1) + 1 / 4 π (2) Of course, both of these equations give only the relation π (2) = 4 15 π (1) To finish, we use π (1) + π (2) = 1 and get π (1) = 15 19 , π (2) = 4 19 Problem 4. Pb 46 There are three choices for state space for this problem: 1) We can chose the states to be the number of umbrellas at his current location (alternating between home and office); or 2) at the office; or 3) at home.
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