Solved: Exponential distribution The occurrence of random

Chapter 13, Problem 66

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Exponential distribution The occurrence of random events (such as phone calls or e-mail messages) is often idealized using an exponential distribution. If l is the average rate of occurrence of such an event, assumed to be constant over time, then the average time between occurrences is l-1 (for example, if phone calls arrive at a rate of l = 2>min, then the mean time between phone calls is l-1 = 12 min). The exponential distribution is given by f 1t2 = le-lt, for 0 t 6 _. a. Suppose you work at a customer service desk and phone calls arrive at an average rate of l1 = 0.8>min (meaning the average time between phone calls is 1>0.8 = 1.25 min). The probability that a phone call arrives during the interval 30, T4 is p1T2 = 1 T 0 l1e-l1t dt. Find the probability that a phone call arrives during the first 45 s (0.75 min) that you work at the desk. b. Now suppose that walk-in customers also arrive at your desk at an average rate of l2 = 0.1>min. The probability that a phone call and a customer arrive during the interval 30, T4 is p1T2 = L T 0 L T 0 l1e-l1tl2e-l2s dt ds. Find the probability that a phone call and a customer arrive during the first 45 s that you work at the desk. c. E-mail messages also arrive at your desk at an average rate of l3 = 0.05>min. The probability that a phone call and a customer and an e-mail message arrive during the interval 30, T4 is p1T2 = L T 0 L T 0 L T 0 l1e-l1tl2e-l2sl3e-l3u dt ds du. Find the probability that a phone call and a customer and an e-mail message arrive during the first 45 s that you work at the desk.