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# hw11_sol - Homework 11 Solutions Exercise 5 page 555 The...

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Homework 11 Solutions April 28, 2009 Exercise 5, page 555 The time between arrivals at the game room in the student union is exponential with mean of 10 minutes. (a) What is the arrival rate per hour? (b) What is the probability that no students will arrive at the game room during the next 15 minutes? (c) What is the probability that at least one student will visit the game room during the next 20 minutes? Solution Let X denote the interarrival time at the game room. We are given that X is exponential with mean of 10 minutes, i.e., X has distribution function f ( x ) = λe - λx with x 0, and the probability that the interarrival random variable X takes value less than a (nonnegative scalar) is given by P { X a } = a 0 6 e - 6 x dx. Evaluating this integral, we have for any a 0: P { X a } = 1 - e - 6 a . (1) We are given that the mean interarrival time is E [ X ] = 10 minutes. We also know that the mean interarrival time and the arrival rate λ are related as follows: E [ X ] = 1 λ . (a) The mean E [ X ] in units of hours is E [ X ] = 1 6 hours. Thus the rate per hour is λ = 1 E [ X ] = 6 . (b) This is equal to the probability that the interarrival time X is greater than 15 minutes, or 1/4 of an hour, which is given by P X 1 4 = 1 - P X 1 4 = 1 - 1 / 4 0 6 e - 6 x dx. 1

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By using relation (1) to evaluate the integral, we obtain P X 1 4 = e - 6 1 4 = e - 3 2 . (c) This is the same as asking for the probability that the interarrival time is 20 minutes, or 1/3 of an hour, which is given by P X 1 3 = 1 - e - 6 1 3 = 1 - e - 2 . Exercise 12, page 556 The U of A runs two bus lines on campus: red and green. The red line serves north campus, and the green line serves south campus with a transfer station linking the two lines. Green buses arrive randomly (exponential inter-arrival time) at the transfer station every 10 minutes. Red buses also arrive randomly every 7 minutes. (a) What is the probability distribution of the waiting time for a student arriving on the red line to get on the green line?
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