hw7soln - ORIE 3500/5500, Fall 10 HW 7 Solutions Homework 7...

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ORIE 3500/5500, Fall ’10 HW 7 Solutions Homework 7 Solutions Problem 1 (a) Let X A ,X B ,X C ,X D be the running times of A, B, C and D. Then the fastest time of team 1 is X 1 = min( X A ,X B ), the fastest time of team 2 is X 2 = min( X C ,X D ) Hence X 1 Exp ( λ A + λ B ) = Exp ( 1 10 + 1 12 ) = Exp ( 11 60 ) X 2 Exp ( λ C + λ D ) = Exp ( 1 8 + 1 14 ) = Exp ( 11 56 ) (b) P ( X 2 = min( X 1 ,X 2 )) = 11 56 11 60 + 11 56 = . 517 Problem 2 Note that if X Binomial ( n,p ) then X = n i =1 Y i where Y i ’s are indepen- dent Bernoulli variables (i.e Y i = 1 with probability p and 0 with probability 1 - p ). M Y i ( t ) = E [ e tY i ] = pe t + (1 - p ) e 0 = pe t + (1 - p ) M X ( t ) = M Y i ( t ) = n Y i =1 M Y i ( t ) = n Y i =1 ( pe t + 1 - p ) = ( pe t + 1 - p ) n Now we take derivatives M 0 ( t ) = n ( pe t + 1 - p ) n - 1 pe t M 00 ( t ) = n ( n - 1)( pe t + 1 - p ) n - 2 ( pe t ) 2 + n ( pe t + 1 - p ) n - 1 pe t E ( X ) = M 0 (0) = n ( p + 1 - p ) n - 1 p = np E ( X 2 ) = M
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This note was uploaded on 11/11/2010 for the course ORIE 3500 at Cornell University (Engineering School).

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hw7soln - ORIE 3500/5500, Fall 10 HW 7 Solutions Homework 7...

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