CS 3341 (Dr. Baron)

Practice Final Exam

1. Consider an M/M/1 queuing system with the average interarrival time of 5 minutes and the average service time of 4 minutes. Compute

(a) the fraction of time when the server is idle;
(b) the fraction of time when there are at least 4 jobs in the system;
(c) expectation of the number of jobs in the system at any time.

2. Lunar eclipse is a rare event that occurs at the average rate of 3 eclipses in 5 months according to a Poisson process.

(a) Compute the probability of exactly 4 lunar eclipses during the year 2002.
(b) Compute the mean and variance for the time until the 50-th lunar eclipse. Compute (or use a suitable approximation) the probability of at least 50 lunar eclipses during a six-year period.

3. Takeoffs and landings at a certain airport are restricted during severe weather conditions. No takeoffs are allowed when lightning is observed around the airport area. Updates are issued every minute. It has been noticed that the probability of lightning is 30% if lightning was observed during the preceding minute. However, this probability is only 10% if there was no lightning a minute ago.

(a) Takeoffs are not allowed at 10 am. Compute the probability that takeoffs will be allowed at 10:03.
(b) Meteorologists predict a 40% chance of lighting at 10 am. Compute the probability that takeoffs will be allowed at 10:03.
(c) Compute the probability that takeoffs will be allowed at 19:33.

4. An electronic parts factory produces resistors. Statistical analysis of the output suggests that the resistances follow an approximately normal distribution with the standard deviation of 0.156 ohms. A sample of 60 resistors has the average resistance of 0.55 ohms.

(a) Based on these data, construct a 95% confidence interval for the population mean resistance.
(b) If the actual population mean resistance is exactly 0.5 ohms, what is the probability that an average of 60 resistances is 0.55 ohms or higher?

5. Based on the data in # 4, is there a significant evidence at a 2% level of significance that the population mean resistance is

(a) greater than 0.5 ohms?
(b) not equal to 0.5 ohms?

6. Consider an electronic message center that can transfer only one message at a time. Also, it has sufficient memory for only 1 other message. If a message arrives while the memory is full, this message will be returned. Messages arrive according to a Binomial counting process with 2-second frames and the average arrival rate of 3 messages per minute. The average time it takes to transfer one message is 10 seconds.

(a) Find the steady-state distribution of the number of messages in the message center.
(b) Compute the average number of messages in the center at any time.

7. The Southeastnorthwest Telephone Company models long-distance calls of its customers by a Binomial counting process. On the average, customers place calls every 2 seconds. The probability of a call during any given frame is p.

(a) What frame length provides p=0.05?
(b) Using this frame length, compute the probability of more than 1,700 calls between 6 pm and 7 pm. Use the Normal approximation of the Binomial distribution.

For the optimal allocation of disk space to hold a large number of images, five digital images are randomly selected for statistical analysis, with the following results:

Image	Size (Mb)
1	120
2	100
3	100
4	200
5	80

(a) Assuming normal distribution of sizes, construct a 90% confidence interval for the mean image size.
(b) Is there significant evidence, at a 10% level of significance, that the mean size is greater than 100 Megabytes?

The following is a random sample from Uniform(a,b) distribution 6, 4, 5, 2, 4, 0, 5, 3, 6, 5
(a) Estimate a and b by the method of moments.
(b) Estimate a and b by the method of maximum likelihood.

10.

A customer service representative answers telephone calls from customers in the order they are received. On the average, customers call every 15 minutes. After the call is answered, it takes an average of 10 minutes to complete it. The system does not allow customers to be "on hold", thus the system's capacity is limited by 1 job. If this system is modeled by a Bernoulli single-server queuing process with limited capacity and (1/2)-minute frames,

(a): Find the steady-state distribution of the number customers is the system.
(b): Compute the expected number of customers in the system at any time.

11.

We have to accept or reject a large shipment of items. For quality control purposes, we collect a sample of 200 items and find 24 defective items in it.

(a): Construct a 96% confidence interval for the proportion of defective items in the whole shipment.
(b): The manufacturer claims that at most one in 10 items in the shipment is defective. At the 4% level of significance, do we have sufficient evidence to disprove this claim? Do we have it at the 3% level?

SOLUTIONS

Michael Baron