Simulation
Final Exam, 2005
Selected entries from
the standard normal table: (90%,95%,97.5%) = (1.28, 1.645, 1.96). For all
problems that require random values, you are to use the following random
numbers, in order. The symbol U denotes a uniform [0,1) random number.
Problems 6 and 8 count 20 points, the rest count 10 each.
ln(U_{1}) 
.32 
.64 
.81 
.57 
.94 
.80 
1.64 
1.28 
1.04 






U 
.73 
.83 
.95 
.12 
.31 
.30 
.67 
.54 
.41 
.06 
.85 
.35 
.30 
.46 
.01 
1. You have observed 50 independent samples from
an unknown distribution F. The sample
mean is 48.4 and S^{2} is 14.6.
How many additional observations would you estimate are required in
order for the length of the 95% confidence interval for the mean of the
distribution to be no greater than 1? (175)
2. You are
simulating a random variable X from the probability density function f(x) = 4x/9
– x^{2}/9, 0 ≤ x
≤ 3. Using the Rejection
method with the Uniform distribution for g(x), calculate the first simulated
value of X. [Note: the maximum value of
f(x) occurs when x=2.] (2.19)
3. You are given the linear congruential random
number generator x_{n+1} = 5x_{n}
+ 6 (mod 19). Using the seed x_{0} = 3, generate 4 random numbers. Use
the 4 numbers to estimate _{}. (3.79)
4. You are
trying to estimate the average grade on the Final Exam for a certain class, and
have observed the following sample of 2 student grades: X_{1},X_{2}
= 66,72. You decide to use the estimator
(X_{1} + 2X_{2})/3 for the mean. Calculate the bootstrap estimate of the mean
square error for your estimate of the mean. (5)
5. You are
testing the hypothesis that the following sample 0.60, 1.12, 0.84, 0.04 is from
an exponential distribution with mean 1.
What is the value of the KolmogorovSmirnov test statistic D? (.326)
6. You have observed the following 2 samples
from unknown distributions:
X_{1}=4, X_{2}=2.6, X_{3}=1,
X_{4}=2; Y_{1}=2.5, Y_{2}=3.
(a)
If you use the twosample test to test the null hypothesis that both samples
were drawn from the same distribution, what is the exact pvalue you obtain? (4/5)
(b)
What is the approximate pvalue for this test in terms of the standard
normal random variable Z? Would you
accept or reject the null hypothesis? (.64, accept)
(c) A third sample, W_{1} = 6, W_{2}
= 2, W_{3} = 1 is taken. Calculate the multisample Kruskal  Wallis
statistic used to test whether or not three samples all come from the same
distribution. (.344)
7.
Use thinning to simulate efficiently the nonhomogeneous Poisson Process with
intensity function l(t) given below. After thinning, what are
the times of arrival in the first two hours?
(Use random number table at top of exam)

ì 
(t+1)
/2 
(t in hours). (0.96) 

l(t) 
=
í 




î 
2
 t 
1 < t < 2 
8. You are interested in
buying a European Call option on a stock whose price is currently $10. The
length of the option is two weeks and the striking price is $11. The price of
the stock was 9 two weeks ago and 8 one week ago. You model the price S_{n}
of the stock in n weeks by S_{n} = S_{0}e^{(X}_{1}^{+…+X}_{n}^{)},
where the X_{i} are independent normal random variables with mean m and variance s^{2}.
(a) Estimate the parameters m and s^{2}. (.0527,
.058)
(b) Use the Polar method to
generate 2 independent normal variables with parameters given by your estimates
in (a). (.181, .237)
(c) Use the answer in (b) to
simulate a value of S_{2}, the price in two weeks. (15.20)
(d) Suppose you simulate a
second value S_{2} = $10.50. Use
this value and your answer in (c) to estimate a fair price for the Call option.
Assume the interest rate is 0. (2.10)