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people.brandeis.edu:~igusa:Math56aS08:oldHW02

# people.brandeis.edu:~igusa:Math56aS08:oldHW02 - MATH 56A...

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MATH 56A: FALL 2006 HOMEWORK AND ANSWERS 2. Math 56a: Homework 2 p. 35 #1.1, 1.2, 1.3 1.1. Every morning a newspaper is added to a pile. With probability 1 / 3 the pile is emptied emptied. But if the pile has 5 newspapers, it is always emptied. Make this into a Markov chain. This is a Markov process with states 0 , 1 , 2 , 3 , 4 representing the number of newspapers in the pile in the evening. The transition matrix is P = 1 / 3 2 / 3 0 0 0 1 / 3 0 2 / 3 0 0 1 / 3 0 0 2 / 3 0 1 / 3 0 0 0 2 / 3 1 0 0 0 0 1.2. Given that P = 1 / 3 2 / 3 3 / 4 1 / 4 What is the probability that X 3 = 1 given that X 0 = 0? The answer is the (0 , 1) entry of P 3 . Instead of doing this in the straightforward boring method I will use right eigenvectors: Pv i = λ i v i . The eigenvalues of P are 1 , 5 / 12 with right eigenvectors v 0 = (1 , 1) and v 1 = (8 , 9): e 0 = 9 17 v 0 + 1 17 v 1 P 3 e 0 = 9 17 v 0 + 5 12 3 1 17 v 1 whose 0th coordinate is p 3 (0 , 1) = 9 17 + 5 12 3 8 17 = 109 216 1

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2 HW AND ANSWERS 2006 1.3. Given that P = . 4 . 2 . 4 . 6 0 . 4 . 2 . 5 . 3 what is the long term probability of being in state 1?
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