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# hw8sln - = P X i 1 = 2 | X i = 1 Y i = H P Y i = H | X i =...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 8 Solution Chapter 4. 2 Since we need to track the past three day’s whether, and each day’s whether can be either Rain (R) or Dry (D), we need 8 states. 3 The transition matrix is as follows: DDD DDR DRD DRR RDD RDR RRD RRR DDD 0.8 0.2 0 0 0 0 0 0 DDR 0 0 0.2 0.8 0 0 0 0 DRD 0 0 0 0 0.6 0.4 0 0 DRR 0 0 0 0 0 0 0.4 0.6 RDD 0.6 0.4 0 0 0 0 0 0 RDR 0 0 0.4 0.6 0 0 0 0 RRD 0 0 0 0 0.6 0.4 0 0 RRR 0 0 0 0 0 0 0.4 0.6 S 1 S 2 S 3 in the column denotes respectively the weather 2 days before yesterday, the weather the day before yesterday, and the weather yesterday. S 1 S 2 S 3 in the row de- notes respectively the weather the day before yesterday, the weather yesterday, and the weather today. 5 P 3 = 0 1 2 0 13/36 11/54 47/108 1 4/9 4/27 11/27 2 5/12 2/9 13/36 P { X 3 = 1 } = 1 4 · 11 54 + 1 4 · 4 27 + 1 2 · 2 9 = 169 432 = 43 216 P { X 3 = 2 } = 1 4 · 47 108 + 1 4 · 11 27 + 1 2 · 13 36 = 169 432 E ( X 3 ) = 1 · 43 216 + 2 · 169 432 = 53 54 1

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8 Let the state X i on any day i be the number of the coin that is ﬂipped on that day, let Y i be the result of the ﬂip on day i . Then P 11 = P [ X i +1 = 1 | X i = 1] = P [ X i +1 = 1 | X i = 1 , Y i = H ] P [ Y i = H | X i = 1] + P [ X i +1 = 1 | X i = 1 , Y i = T ] P [ Y i = T | X i = 1] = 1 × 0 . 7 + 0 = 0 . 7 P 12 = P [ X i +1 = 2 | X i = 1]
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Unformatted text preview: = P [ X i +1 = 2 | X i = 1 , Y i = H ] P [ Y i = H | X i = 1] + P [ X i +1 = 2 | X i = 1 , Y i = T ] P [ Y i = T | X i = 1] = 0 + 1 × . 3 = 0 . 3 P 21 = P [ X i +1 = 1 | X i = 2] = P [ X i +1 = 1 | X i = 2 , Y i = H ] P [ Y i = H | X i = 2] + P [ X i +1 = 1 | X i = 2 , Y i = T ] P [ Y i = T | X i = 1] = 1 × . 6 + 0 = 0 . 6 P 22 = P [ X i +1 = 2 | X i = 2] = P [ X i +1 = 2 | X i = 2 , Y i = H ] P [ Y i = H | X i = 2] + P [ X i +1 = 2 | X i = 2 , Y i = T ] P [ Y i = T | X i = 2] = 0 + 1 × . 4 = 0 . 4 The transition matrices are P = ± . 7 0 . 3 . 6 0 . 4 ² P 3 = ± . 667 0 . 333 . 666 0 . 334 ² The probability that coin 1 is ﬂipped on the third day is 1 2 ( P 3 11 + P 3 21 ) = 0 . 666. 2...
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hw8sln - = P X i 1 = 2 | X i = 1 Y i = H P Y i = H | X i =...

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