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(U1) 0.32 0.64 0.81 0.57 0.94 0.8 1.64 1.28 1.04 U 0.73 0.83 0.95 0.12 0.31 0.3 0.67 0.54 0.41 .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 S2 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 – x2/9,   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  xn+1 = 5xn + 6 (mod 19). Using the seed x0 = 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: X1,X2 = 66,72.  You decide to use the estimator (X1 + 2X2)/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 Kolmogorov-Smirnov test statistic D?  (.326)

6.   You have observed the following 2 samples from unknown distributions:

X1=4, X2=2.6, X3=1, X4=2;       Y1=2.5, Y2=3.

(a) If you use the two-sample test to test the null hypothesis that both samples were drawn from the same distribution, what is the exact p-value you obtain?       (4/5)

(b)  What is the approximate p-value 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, W1 = 6, W2 = 2, W3 = 1 is taken. Calculate the multi-sample 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 non-homogeneous 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)

0 < t < 1

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 Sn of the stock in n weeks by Sn = S0e(X1+…+Xn), where the Xi are independent normal random variables with mean m and variance s2.

(a)    Estimate the parameters m and s2.  (.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 S2, the price in two weeks. (15.20)

(d)    Suppose you simulate a second value S2  = \$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)