45It can be shown that if events are occurring in time according to a Poisson distribution with mean λt, then the interarrival times between events have an1λexponential distribution with meanSuppose that customers arrive at a checkout counter at the rate of two per minute.(a) What are the mean (in minutes) and variance of the waiting times between successive customer arrivals?minmeanvariance(b) If a clerk takes 2.7 minutes to serve the first customer arriving at the counter, what is the probability that at least one more customer will be waiting whenthe service to the first customer is completed? (Round your answer to four decimal places.)

a. The mean of the interarrival times is (λ1=21)minutes.The variance of the interarrival times is…

Answered: 45 It can be shown that if events are…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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What is the probability that the total of two randomly chosen numbers exceeds 3, where the first number is chosen from a uniform distribution between 0 and 2, and the second number is selected from an exponential distribution with a mean of 1?
3 Historical evidence indicates that times between fatal accidents on scheduled American domestic passenger flights have an approximately exponential distribution. Assume that the mean time between accidents is 44 days. (a) If one of the accidents occurred at 12:00 am on August 1 of a randomly selected year in the study period, what is the probability that another accident occurred that same month? (Round your answer to four decimal places.) (b) What is the variance of the times between accidents?
6. Consider a coffee shop that uses a single-server queue. The inter-arrival time is exponentially distributed with a mean of 10 minutes and the service time is also exponentially distributed with a mean of 8 minutes. Calculate the: (i) mean wait in the queue (2) (ii) mean number in the queue (2) (iii) the mean wait in the system (2) (iv) mean number in the system (2) (v) proportion of time the server is idle. (2) [10] 7. The arrival of customers at a bank follows a Poisson distribution with a mean arrival rate of 10 customers per hour. The service time at the bank follows an exponential distribution with a mean service rate of 6 minutes per customer. Calculate the average number of customers in the system (including those being served and waiting) and the average time a customer spends in the system. How is the system performing? [8]
5. Suppose the time spent by a randomly selected student at a campus computer laboratory has a gamma distribution with mean 20 min and variance 80 min². (a) What are the values of a and B? (b) What is the probability that a student spends between 20 and 40 min at the laboratory? (c) Only 1% of students will spend more than how many minutes at the laboratory?
Suppose that an individual stock’s return is normally distributed with a mean of 9% and a standard deviation of 4%. What is the probability that the stock’s return will be less than 12%? (please round your answer to 4 decimal places)
7. Sunshine Blu-ray players have a funtional life that is nomally distributed with mean of 2.2 and a standard deviation of 0.3 year. What is the probability that a Blu-ray player years will break down during the first two years?
(c) Does this Poisson distribution provide a good model for the number of accidents in this example (as per Table 1)? Explain why or why not. Use R to produce a graph to help support your answer (produce the R as part of your answer). (d) Use the model in (b) to estimate the probability that less than 2 accidents were recorded in any randomly chosen week. How does this compare to the actual data?
Customers arrive to a local bakery with an average time between arrivals of5 minutes. However, there is quite a lot of variability in the customers’ arrivals, asone would expect in an unscheduled system. The single bakery server requires anamount of time having the exponential distribution with mean 4.5 minutes to servecustomers (in the order in which they arrive). No customers leave without service.f. Why are the estimated waits in this system so long? Are the assumptions behindthem reasonable? Why or why not?

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