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Unformatted text preview: Homework Set #5 Solutions IE 336 Spring 2011 1. (a) p , q , and r can be determined by solving the matrix form of the system of equations: 1 2 2 2 4 / 3 4 / 3 1 / 2 1 3 p q r = 1 1 1 Solving yields p = 1 / 4, q = 1 / 8, and r = 1 / 4. (b) Using the values of p , q , and r from part (a), the transition matrix can be rewritten as follows: P = 1 / 2 1 / 4 1 / 4 1 / 6 1 / 2 1 / 3 3 / 8 3 / 4 1 / 8 The walk probability is: p 313112232322312312 = p 4 31 p 13 p 11 p 3 12 p 2 22 p 4 23 p 2 32 = 8 . 2784 10 10 (c) p (5) 21 = 0 . 2328 2. (a) The probability that the process is in any of the three states at some point k is equally likely, i.e., P ( X k = 1) = P ( X k = 2) = P ( X k = 3) = 1 / 3. The probability that the process will be in state 3 four steps later is: P ( X k +4 = 3) = 3 X i =1 P ( X k +4 = 3  X k = i ) P ( X k = i ) = 3 X i =1 p (4) i 3 P ( X k = i ) Let p k = P ( X k = 1) P ( X k = 2) P ( X k = 3) . Using matrix notation:= 3) ....
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This note was uploaded on 01/19/2012 for the course IE 230 taught by Professor Xangi during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 Xangi

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