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hw6sln

# hw6sln - IEOR 161 Operations Research II University of...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 6 Suggested Solution Chapter 5. 58. Let X j be the time it takes for the first type j coupon collected. P { i is the last type collected } = P { X i = max j =1 ,...,n X j } = 0 ( X j < x j = i ) λ i e - λ i x dx = 0 j = i (1 - e - λ j x ) λ i e - λ i x dx = 1 0 j = i (1 - y - λ j i ) dy ( y = e - λ i x ) = E j = i (1 - U λ j i ) 68. (a) E [ A ( t ) | N ( t ) = n ] = E n i =1 A i e - α ( t - S i ) = E [ A ] e - αt E n i =1 e αU [ i ] = E [ A ] e - αt E n i =1 e αU i = nE [ A ] e - αt E e αU = nE [ A ] e - αt t 0 e αx 1 t dx = nE [ A ] 1 - e - αt αt E [ A ( t )] = E [ E [ A ( t ) | N ( t ) = n ]] = E [ N ( t )] E [ A ] 1 - e - αt αt = E [ A ] 1 - e - αt αt λt = E [ A ] λ (1 - e - αt ) α (b) Going backwards from time t to 0, events occur according to a Poisson process and an event occurring at time s has value Ae - αs attached to it. 72. (a) Define the random variable S n as the departure time of the last rider. Since it is the sum of n independent exponentials with rate λ , it has a gamma distribution with parameters n and λ .

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hw6sln - IEOR 161 Operations Research II University of...

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