hw9_sol - IEOR 172: Probability and Risk Analysis for...

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IEOR 172: Probability and Risk Analysis for Engineers, Fall 2007 Homework 9 Solution Chapter 7 Question 11 Let X i equal 1 if a changeover occurs on the i th flip and 0 otherwise. Then E[ X i ] = P { i - 1 is H,i is T } + P { i - 1 is T,i is H } = 2(1 - p ) p, i 2 . E[number of changeovers] = E " n X i =1 X i # = n X i =1 E[ X i ] = 2 p ( n - 1)(1 - p ) . Question 22 From example 3f, 1+6/5+6/4+6/3+6/2+6. Question 26 (a) E[max] = Z 1 0 P { max > t } dt = Z 1 0 (1 - P { max t } ) dt = Z 1 0 (1 - t n ) dt = n n + 1 . (b) E[min] = Z 1 0 P { min > t } dt = Z 1 0 (1 - t ) n dt = 1 n + 1 . Question 32 Use the notation in Problem 9, X = n X j =1 X j where X j is 1 if box j is empty and 0 otherwise. Now, with E[ X j ] = P { X j = 1 } = n X i = j (1 - 1 /i ) , we have that Var( X j ) = E[ X j ](1 - E[ X j ]) . 1
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Also, for j < k , Cov( X j ,X k ) = k - 1 X i = j (1 - 1 /i ) n X i = k (1 - 2 /i ) - n X i = j (1 - 1 /i ) n X i = k (1 - 1 /i ) Var( X ) = n X j =1 E[ X j ](1 - E[ X j ]) + 2Cov( X j ,X k ) . Question 34
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hw9_sol - IEOR 172: Probability and Risk Analysis for...

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