Homework_4

Homework_4 - 1. Read pages 192, 193 of the textbook. 2....

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1. Read pages 192, 193 of the textbook. 2. Read the remark on page 200 of the textbook. 3. Read Proposition 4.3. 4. [15 points] A stochastic process { X n ,n = 0 , 1 , 2 , ···} with state space S = { 0 , 1 , 2 , ··· ,l } is called a Markov chain if for any i,j,i 0 ,i 1 , ··· ,i n - 1 S and n 0, P { X n +1 = j | X n = i,X n - 1 = i n - 1 , ··· ,X 1 = i 1 ,X 0 = i 0 } = P { X 1 = j | X 0 = i } . a) Use this definition to prove that for the Markov chain defined as above, and for n > k 0 P { X n = j | X k = i,X k - 1 6 = i k - 1 , ··· ,X 1 6 = i 1 ,X 0 6 = i 0 } = P { X n - k = j | X 0 = i } . b) For the Markov chain defined as above and for n > k 0, does P { X n = j | X k 6 = i,X k - 1 = i k - 1 , ··· ,X 1 = i 1 ,X 0 = i 0 } = P { X n - k = j | X 0 6 = i } ? Prove it or disprove it. 5. [20 points] Consider a Markov chain with 4 states { 0 , 1 , 2 , 3 } . The transition proba- bility matrix takes the following form: 2 / 5 1 / 5 1 / 5 a 1 / 5 2 / 5 1 / 5 b 1 / 5 1 / 5 2 / 5
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This note was uploaded on 01/02/2012 for the course MATH 2603 taught by Professor Han during the Spring '10 term at HKU.

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Homework_4 - 1. Read pages 192, 193 of the textbook. 2....

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