IE 581 Introduction to Stochastic Simulation. One page of notes, front and back. Closed book. 50 minutes. Score

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1 One page of notes, front and back. Closed book. 50 minutes. Score Schmeiser Page 1 of 4 Test #1, Spring 2001

2 1. True or false. (If you wish, write an explanation of your thinking.) (a) T Data are "binary" if each observation is a zero or a one. (b) F If θˆ = and ste ˆ(θˆ) = , then the digit "3" in θˆ meaningless. is (c) F Increasing the sample size by a factor of ten yields one additional meaningful digit in the point estimator. (d) F If random numbers are uniformly distributed in one dimension, then they are also uniformly distributed in two dimensions. (e) F The numbers 5 and 15 are relatively prime because all of their factors are prime. (g) T The binary number 1111 is in decimal. (h) F The cumulative distribution function (cdf) of a continuous random variable X evaluated at X = 0.5 is uniformly distributed on (0, 1). (i) F The MSExcel command "=if(rand()<.6,1,0)" is an example of the inverse transformation (to generate a "1" with probability 0.6). (j) F When generating random variates using the inverse transformation F X 1 (u ), the cdf F X must be closed form. (k) F When using a linear congruential random-number generator = (a 1 + c ) (mod m ), the largest possible value of is m. 2. Consider the random-number generator based on the linear congruential relationship = 6 1 (mod 7). (a) If X 6 = 5, what are the possible values of X 5? There are only six possible values of Therefore, if X 6 = 5, then X 5 = 2 (b) If we begin with the initial random-number seed X 0, will Floyd s algorithm have warned before seed X 10 is used? (a) yes (b) no (c) depends on the value of X 0. Schmeiser Page 1 of 4 Test #1, Spring 2001

3 3. Monte Carlo simulation can be used to evaluate an integral by interpreting it as an expected value E[g (x )] = g (x ) f X (x ) dx, where f X is the density function of X. Consider the integral which can be simulated using sum = 0 for i=1 to 1000 { u = rand() x=10*u sum=sum+10*x*x } thetahat = sum / x 2 dx, (a) In this code, what is the function g (x )? g (x ) = 10 x 2 (b) In this code, what is the density function f X? s uniformly distributed over (0, 10). Therefore, f X (x ) = 1 / 10 for 0 x 10 and zero elsewhere. (c) In this code, the point estimator is (choose one) (a) thetahat (b) 973 / 3 (c) unspecified. Schmeiser Page 2 of 4 Test #1, Spring 2001

4 4. Suppose that you are using Monte Carlo simulation to estimate a probability p using pˆ, the average of n independent binary observations. The variance of pˆ is [p (1 p )] /n. (a) Suppose that the value of p is quite close to zero. Such simulations are time consuming because, on the average, 1 /p trials are necessary to obtain one success; most of the computer time is spent generating failures. A colleague suggests reducing variance by estimating q = 1 p, using the much larger qˆ = " number of failures" /n. Is this a good idea? Discuss briefly. No. The variance of qˆ = 1 pˆ is the same same for pˆ. (b) For a given sample size n, the largest variance occurs for p = 0.5. Yet "rare-event" simulations, where p is close to zero or one, are considered more difficult than the p = 0.5 case. Explain why? Here is one of several reasonable arguments. To simplify, assume that p is close to zero, such as p = Although the variance is largest when p = 1 / 2 and smaller when p is close to zero, the standard error of pˆ needs to be quite a bit smaller than p for the point estimator to be useful. Suppose that we want the standard error to be 10% of p (which for small values of p essentially guarantees that the left-most non-zero digit of pˆ will round to a value close to the correct digit.) Then.1 = ste(pˆ ) = p (1 p ) /n 1 / (pn ), p p which implies that n 100 /p. Therefore, p = 1 / 2 requires n 200 and, for example, p = 10 8 requires n 10 billion. 5. Consider Machine A, which fails when either of two independent components fails. Consider also Machine B, which is identical to Machine A except that it fails when both components fail. Let µ i denote the expected failure time for Machine i, for i = A, B. Let X A j and X B j, for j = 1, 2, denote the failure times of Component j for Machines A and B, respectively. (For example, X A 1 is the failure time of Component 1 of Machine A.) Suppose that we wish to estimate µ A µ B, the difference in mean failure times, using common random numbers. (a) What is the relationship between X 1 A and X 1 B? equal independent not enough information to know (b) What is the relationship between X 1 A and X 2 A. equal independent not enough information to know Schmeiser Page 3 of 4 Test #1, Spring 2001

5 6. Given a random-number u, explain how to toss a six-sided die. There are many solutions. An easy one is the inverse transformation u = random number x = 6u + 1 if (x>6), then x = 6, where x is the value of the die toss and y denotes the integer part of y. (The "if" statement is needed when u rounds to one.) 7. Consider the sentences "The point estimator Y of the performance measure E(Y ) has standard error ste(y) = std(y i ) / n if the output data Y 1, Y 2,...,Y n are independent and identically distributed (iid). The standard-error estimator is std ˆ (Y) = S/ n, where S 2 n = [Σi =1 Y 2 i n (Y) 2 ] / (n 1). Suppose that you rerun a simulation experiment, keeping everything the same except the random-number seeds. Using the above two sentences for context, circle "constant", "random", and "undefined" for each part below. (a) Y constant random undefined (b) E(Y ) constant random undefined (c) ste(y) constant random undefined (d) Y 2 constant random undefined (e) std ˆ (Y) constant random undefined (f) n constant random undefined (g) Y 1 2 constant random undefined Schmeiser Page 4 of 4 Test #1, Spring 2001