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(U_{1})

.33

.24

.52

1.22

.66

.75




U_{2}

.12

.61

.34

.45

.90

.89





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?

Customers arrive at a single server queueing system according to a nonhomogeneous
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.

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.

The sample mean after 29 independent samples of a distribution is 25
and the sample variance is 18. If the 30^{th} 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)

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

You wish to test if the sample 0, 2 is from a Binomial(2,p) distribution.
What is the pvalue for your test in terms of the chisquare distribution?

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

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 pvalue 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

To test that the arrival process is a nonhomogeneous 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?

After accepting the hypothesis that the number of arrivals per hour
is Poisson, you test that the arrivals each hour follow the same nonhomogeneous
Poisson process. What is the (approximate) pvalue you obtain (in terms
of a chisquare distribution)?