FINAL   MATH 286/366

Spring, 2002

The following table is to be used for each problem requiring random numbers. For each problem, start at the left and use the numbers in order.

 -ln(U1) 0.33 0.24 0.52 1.22 0.66 0.75 U2 0.12 0.61 0.34 0.45 0.9 0.89

1. You are to use the rejection method to sample the function with pdf    f(x) = (2x + 1)/6, 0<x<2. Generate two (accepted) samples from f(x) using this method. Explain your choice of g(x). What is the mean of your two samples?

1. Customers arrive at a single server queueing system according to a non-homogeneous Poisson process with l(t) = 1 + 2t, 0<t<1 (t in hours). The service time is a random variable that is Uniform(0,1). Use the thinning method and the random numbers in the table to perform a simulation. How long does the second customer spend in the system? Generate arrival time, thinning, then service time for each customer in order.

1. Starting at 1:00 PM, students are observed arriving at a bank with two ATM machines according to a Poisson process with mean 6 per minute. They go to whichever machine is free, choosing Machine 2 if both are available. Machine 1 has an exponential service time with mean 1/2 minute and Machine 2 an exponential service time with mean 1/3 minute. Run a simulation to determine when the third customer completes service. Use the table on page 1 to generate arrival time then service time for each customer in order.

1. The sample mean after 29 independent samples of a distribution is 25 and the sample variance is 18. If the 30th sample value is 30, determine a 95% confidence interval for your estimate of the mean of the distribution.  The percentiles from the normal table are (90%,95%,97.5%) = (1.28,1,645,1.96)

1. A sample from an unknown distribution produces X1 = 2 and X2 = 6. Determine the bootstrap approximation to the mean square error involved in estimating the mean with the sample mean.

1. You wish to test if the sample 0, 2 is from a Binomial(2,p) distribution. What is the p-value for your test in terms of the chi-square distribution?

1. You are to test if the sample 1, 3.5, 4.5, 5 is from the uniform distribution on (0,6) using the Komolgorov-Smirnov statistic. What value of D do you obtain?

1. You wish to test if the following two samples come from the same distribution using the rank sum statistic. Sample I: 2, 3, 3.5. Sample II: 1,4,5. What is the exact p-value you obtain?

The following paragraph is to be used in problems 9 and 10. Data is collected from an ATM machine on campus for three different hours during the day. The arrival times of customers, in minutes from the start of the hour, are given below.

Hour 1:  3, 15, 20, 32

Hour 2:  7, 14, 28

Hour 3:  10, 11, 13, 22, 23

1. To test that the arrival process is a non-homogeneous Poisson process for each hour, you first test that the number of arrivals each hour is independent and has the same Poisson distribution. Your test uses the fact that the mean and variance of a Poisson distribution are equal. What is the value of the statistic you obtain?

1. After accepting the hypothesis that the number of arrivals per hour is Poisson, you test that the arrivals each hour follow the same non-homogeneous Poisson process. What is the (approximate) p-value you obtain (in terms of a chi-square distribution)?