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Homework4_Solution

# Homework4_Solution - 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. Solution: a) P { X n = j | X k = i,X k - 1 6 = i k - 1 , ··· ,X 1 6 = i 1 ,X 0 6 = i 0 } = j n - 1 , ··· ,j k +1 ,j k - 1 6 = i k - 1 , ··· ,j 0 6 = i 0 P { X n = j,X n - 1 = j n - 1 , ··· ,X k +1 = j k +1 ,X k = i,X k - 1 = j k - 1 , ··· ,X 0 = j 0 } P { X k = i,X k - 1 6 = i k - 1 , ··· ,X 0 6 = i 0 } = j n - 1 , ··· ,j k +1 ,j k - 1 6 = i k - 1 , ··· ,j 0 6 = i 0 P { X n = j,X n - 1 = j n - 1 , ··· ,X k +1 = j k +1 | X k = i } P { X k = i,X k - 1 = j k - 1 , ··· ,X 0 = j 0 } P { X k = i,X k - 1 6 = i k - 1 , ··· ,X 0 6 = i 0 } = j n - 1 , ··· ,j k +1 P { X n = j,X n - 1 = j n - 1 , ··· ,X k +1 = j k +1 | X k = i } j k - 1 6 = i k - 1 , ··· ,j 0 6 = i 0 P { X k = i,X k - 1 = j k - 1 , ··· ,X 0 = j 0 } P { X k = i,X k - 1 6 = i k - 1 , ··· ,X 0 6 = i 0 } = X

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Homework4_Solution - 1 Read pages 192 193 of the textbook 2...

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