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# hw7 - ORIE 3500/5500 Engineering Probability and Statistics...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 7 Problem 1 Four runners, A, B, C and D are organized in two teams, with runners A and B forming Team 1, and runners C and D forming Team 2. All four runners will run the same distance. Suppose that the running times of the 4 runners are all exponentially distributed, with respective means μ A = 10, μ B = 12, μ C = 8 and μ D = 14 minutes. Assume also that all four running times are independent. ( a ) What is the distribution of the actual fastest time in each team? ( b ) What is the probability that the winner of the race (the fastest one of the four runners) will come from Team 2? Problem 2 Derive the moment generating function of the binomial distribution and use it to derive the mean and the variance of a binomial random variable. Problem 3 If X is a Poisson random variable with mean λ , show that p X ( i ) first increases and then decreases as i increases, reaching is maximum when i is the largest integer less than or equal to λ . Problem 4 You arrive at a bus stop at 10 0’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and

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