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Practice Midterm1

# Practice Midterm1 - IEOR 4106 Midterm Exam Open text book...

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IEOR 4106 Midterm Exam. Open text book and class notes; 1.5 hours. 100 Points total 1. (35 points) Voice messages are made from a cell phone according to a Poisson process at rate 8 per hour, and independent of this, text messages are sent from the phone according to a Poisson process at rate 2 per hour. (Time is in hours.) (a) (10 points) Given that 4 text messages were sent during the hours of 10:00AM to 11:30AM, what is the probability that exactly two of these four were made between 11:00AM and 11:30AM? SOLUTION: We can treat the (unordered) four arrival times U 1 , . . . U 4 as iid uni- formly distributed over the 90 minute interval, which for simplicity we denote by (0 , 90). Each one, independently, would arrive between 11:00AM and 11:30AM with “success” probability p = P ( U (60 , 90)) = 30 / 90 = 1 / 3. So, letting N denote the number of these 4 that do so yields a binomial ( p, 4) distribution; P ( N = 2) = ( 4 2 ) p 2 (1 - p ) 2 = 6(1 / 3) 2 (2 / 3) 2 = 8 / 27. (b) (10 points, 5 each) i. What is the probability that the first two messages are both voice, and the third is text? SOLUTION: lambda 1 = 8 , λ 2 = 2. Letting p = λ 1 / ( λ 1 + λ 2 ) = 0 . 80, we get p 2 (1 - p ) = (0 . 8) 2 (0 . 20) = 0 . 128. ii. Starting from now (time 0) what is the expected length of time until at least one voice and one text message will been made?

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Practice Midterm1 - IEOR 4106 Midterm Exam Open text book...

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