HW7Soln - IEOR 161 - Introduction to Stochastic Processes...

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IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW7 Solutions ** Note that the numbering is from Ross 9th Edition 5.67 If we count a satellite if it is launched before time s but remains in operation at time t , then the number of items counted is Poisson with mean m ( s ) = λ R s 0 G ( t - y ) dy . The answer is e - m ( s ) . 5.72 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 . b) By Theorem 5.2 in textbook, given S n = t the set of times at which the first n - 1 riders departed are independent uniform (0 ,t ) random variables. Therefore, if the rider departs the bus at time s , then he will get home before time t with probability P( s ) = P { Walking time < t - s } = 1 - e - μ ( t - s ) Then, each of these riders will be at home at time t with probability p = Z t 0 P( s ) 1 t ds = Z t 0 ( 1 - e - μ ( t - s ) ) 1 t ds.
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HW7Soln - IEOR 161 - Introduction to Stochastic Processes...

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