632 Introduction to Stochastic Processes Spring 2004 Midterm Exam II Instructions: Show calculations and give concise justiﬁcations for full credit. Points add up 100. 1. (20 pts) Eileen is catching ﬁsh at the Poisson rate of λ per hour. Each ﬁsh is a salmon with probability 1/3 and a trout with probability 2/3. Eileen ﬁshes for 1 hour. Find the probability that she catches exactly 2 trout and at least 1 salmon. (Try to give the simplest expression you can.) 2. (20 pts) Given that a rate λ Poisson process has exactly two arrivals in time interval [0 , t ], compute the mean of the second (later) arrival time. In symbols, ﬁnd the expectation E [ T 2 | N ( t ) = 2]. 3. (15 pts) Consider an M/M/1 queue where customers arrive as a Poisson process with rate λ and services happen at rate μ . The system starts empty, and then customers start arriving one by one. Find the probability that the ﬁrst customer is still in service when the second customer arrives in the system. 4. (15 pts) Consider an M/M/1 queue where customers arrive as a Poisson
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