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part9 - Probability Homework Solution#4 2.71 P[3 or more...

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1 Probability Homework Solution #4 2.71 2 100 3 0 100 99 98 2 4 [3 or more errors] 1 [2 or fewer errors] 100 1 (1 ) 10 100 99 1 (1 ) 100(1 ) (1 ) 2 1 099985 1.5 10 k k k P P p p p k p p p p p = = = = × = + + = = × 2.75 For an individual chip, the problem that the lifetime exceeds 1 α seconds is: 1 1 1 lifetime P e e p α α α > = = ± If we consider the testing of whether each chip lifetime exceeds 1 α as a sequence of Bennoulli trials, then [ ] 10 1 1 10 5 10 5 ( ) (1 ) 0.289 k k k P k e e k = = = 2.82 (a) 0 1 1 1 (1) , (1) 2 2 p p = = (b) 0 0 1 1 0 1 2 1 ( 1) ( ) ( ) 3 6 1 5 ( 1) ( ) ( ) 3 6 p n p n p n p n p n p n + = + + = + In matrix notation, we have [ ] [ ] 0 1 0 1 2 1 3 3 ( 1), ( 1) ( ), ( ) 1 5 6 6 p n p n p n p n + + = , or ( 1) ( ) p n p n + = Ρ
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2 (c) 1 1 (0) , 2 2 2 1 1 1 5 7 3 3 (1) , , 1 5 2 2 12 12 6 6 2 1 5 7 3 5 3 3 (2) , , 1 5 12 12 8 8 6 6 2 1 3 5 17 31 3 3 (3) , , 1 5 8 8 48 48 6 6 2 1 17 31 3 3 (4) , 1 5 48 48 6 6 p p p p p =
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