Summer 2021 Final Exam

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Attempt History Attempt Time Score LATEST Attempt 1 120 minutes 73 out of 100 Score for this quiz: 73 out of 100 Submitted Aug 5 at 6:13pm This attempt took 120 minutes. - Question 1 3/3pts | Find B [3X — 1]. Correct! b 2/3 Suppose X and Y have joint p.df. f(z,y) =8zyforall0 <y<z < 1. Question 2 Correct! a. Yes b. No https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 Are X and Y dependent random variables? YES or NO? Consider again the joint p.d.f. from the previous question, flz,y) =8zyfor0 <y <z <1 2/20

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 . B Question 3 3/3pts | If X and Y are i.i.d. Nor(1, 2) random variables, find Var(X — Y + 2). b. 0.5 c. 1 d. 2 Correct! e 4 ’7 | Question 4 0/3pts | If Z1,Z3, -+ -, Zay, areiid. Nor(0,1), whatis the distribution of 2n 2 1=1 Zi ? a. Nor(0, 2n) b. Pois(2n) orrect Answer c. x2(2n) ou Answered d. Both (b) and (c) e. All of the above. https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 3/20

10V21, 355 PM ISyE 6644 - Summer 2021 - Final Exam : Smulation - ISYE-6644-A/OAN/OO1 IfU, andU, areiid. Unif(0,1), whatis the distribution of ~0.5In((1 - Uy)'U}| ? a. Exp(0.5) b. Erlang,(2) c. Erlang,(—2) e. Erlang,(—0.5) Question 8 3/3 pts Suppose we are given a choice between two estimators, 7} and 75, and are told that the relative efficiency of T} toT5 is(.8. That is, the Mean Square Error (MSE) ratio between the two estimators is ;—:g%;—’;- = 1.8. WAl Which estimator would you choose, and why? -omect a. Choose T . it has a lower MSE. T c. Choose T3, it has a lower MS 1> SE a. stimator would be an equally acceptible choice. Question 9 3/3pts

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 If X1, X9, X3 arei.id. normal, with X; = 10, X9 = 18, and X3 = 14, what is the maximum likelihood estimator for the variance 02 ? a. 8/3 b. 4 Correct! d.32/3 i ) Question 10 0/3pts Suppose that we take three i.i.d. observations X; = 0.6, X, = 2.4, and X3 = 3, from X ~ Exp(A). Using the maximum likelihood estimate for A, find the MLE of Pr(X < 2). ou Answered orrect Answer r . .~ Question 11 0/3pts | i https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 6/20

10V321, 3.55 PM ISyE 6644 - Summer 2021 - Final Exam : Smulation - ISYE-6644-AJOAN/OD1 Question 13 S/3pts | Which one of the following properties of a Brownian motion process W(t) is incorrect? a. W(3) ~ N(0, 3) b. W(4) — W(2) has the same distribution as W(8) — W(4) c. WI(10) — WI(T) i ndent of W(3) — W(2) d. A Brownian Bridge, B({), is a conditioned BM such that wW(i0) —WwW(l)—0 e. Cov(W(1),W(3)) — | Question 14 0/3pts Let W(!) denote a Brownian motion process at time ! . Let A= fol W(t) dt represent the area under the process from time t = 0 tot = 1. Find the probability that A < 1. c. 0.8413 orrect Answer d 09584 e. 0.9817 hitps-//gatech.nstructure.com/courses/ 187990/guizze</279847 7module_item_id=1796916 820

10/3/21, 3:55 PM Correct! Correct! ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Question 15 3/3pts | Suppose X7, ..., Xig are i.i.d. from an Exp(1) distribution. Use the Central Limit Theorem to find an approximate distribution of the sample mean Xq0 and then calculate Pr( X < 1) for me. a. 0.00 b.0.16 c. 0.50 d. 0.84 e. 1.00 Question 16 3/3 pts TRUE or FALSE? ARENA's Input Analyzer allows the user to determine/estimate the underlying distribution of data, and even creates an ARENA expression for that RV that you can paste directly into the simulation model. Question 17 0/3pts | TRUE or FALSE? In ARENA, a DECIDE module can only make decisions on where to go next based on chance/probabilities. https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 9/20

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 TRUE or FALSE? The Kolmogorov-Smirnov test can be used both to see (i) if data seem to fit to a particular hypothesized distribution, and (ii) if the data are independent. True Correct! False B | Question 21 3/3pts | Suppose we're conducting a x2 goodness-of-fit test to determine whether or not 1500 i.i.d. observations are from a shifted Gamma(a, £, ¢) distribution, where «, ,3, and the shift parameter ¢ must all be estimated. If we divide the observations into k = 7 equal-probability intervals, how many degrees of freedom will our test have? Correct! a3 b. 6 c.7 d. 11 e. 1497 B i Question 22 3/3pts | Suppose we're conducting a X2 goodness-of-fit test to determine whether or not 200 i.i.d. observations are from a Bieber distribution. (The Bieber has a very, very ugly-looking p.d.f.) Suppose, after we divide the https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 11/20

10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 observations into various intervals and deal with the various unknown parameters, it turns out that the test has 5 degrees of freedom. Further suppose that we calculate a g-o-f statistic of X(2) = 7.5. Will we ACCEPT (i.e., fail to reject) or REJECT the null hypothesis of the Bieber distribution? Use level of significance o = 0.05 for your test. Correct! a. Accept (i.e., fail to reject) b. Reject i i Question 23 0/3pts | TRUE or FALSE? Simulation output (e.g., consecutive waiting times) is almost never i.i.d., nor normal. orrect Answer ou Answered True False { R Question 24 3/3pts | TRUE or FALSE? Steady-state analysis should only be performed once simulation output initialization bias effects are dealt with (or at least considered). Correct! True False https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 12/20

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Suppose you are using the method of common random numbers to ascertain which of two different queueing system configurations | or |l results in lower customer waiting times. Which of the following steps can you best use to carry out this task? a. Conduct runs of configurations | and Il independently. b. Use the same customer arrival times for configurations | and 1. c. Use the same customer service times for configurations | and |I. Correct! d. Both (b) and (c). e. None of the steps listed above can be used to carry out this task. Question 28 3/3pts | Which variance reduction technique is most closely associated with a paired-t confidence interval for the mean? a. Antithetic Random Numbers Correct! b. Common Random Numbers c. Control Variates d. Both (a) and (b) e. None of the above I | Question 29 3/3pts | https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 14/20

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 We are studying the waiting times arising from two queueing systems. Suppose we make four independent replications of both systems where the systems are simulated independently of each other. The average waiting times from each replication of the systems are shown in the table below. Assuming that the average waiting time results from each replication are approximately normal, find a two-sided 95% confidence interval for the difference in the means of the two systems. Replication| System 1 | System 2 1 10 25 20 40 2 3 25 30 4 35 20 a. [—52.46, 39.96] b. [—40.09, 27.59] Correctl c. [—22.17,9.67] d. [~2.97,14.67] e. [12.46,27.72] Question 30 0/3pts | Assume we are still studying the waiting times arising from two queueing systems using the same System 1 observations as on the previous problem. But now we have used common random numbers in the System 2 simulation to induce positive correlation between the results of the two systems. Again find a two-sided 95% confidence interval for the difference in the means of the two systems. Replication| System 1 | System 2 https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 15/20

10/3/21, 3:55 PM Correct! Correct! ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Consider a normal ranking and selection problem in which we are trying to determine which of three (simulated) queueing configurations minimizes our expected wait time. After the R&S procedure finishes, we have the following sample average times: X; = 100, X, = 500, and X3 = 300. Which system do you choose as best? a. System 1 b. System 2 c. System 3 Question 33 3/3pts Let's conduct a taste test to determine which of Coke vs. Pepsi vs. Dr. Pepper is Atlanta's most-preferred soft drink. Without going into the details regarding the parameter choices for P* and §*, let's just suppose that the single-stage multinomial ranking-and-selection procedure from class tells us to survey 1000 people. But after just 700 people, suppose that 451 love Coke, while only 150 enjoy Pepsi and only 99 prefer Dr. Pepper. What do you do? a. You are stubborn and inefficient - you take all 1000 samples even though Pepsi and Dr. Pepper cannot possibly catch up. b. You are smart and efficient - since the R&S procedure will select the soft drink based solely on which one gets more wins, you stop now, select Coke as the winner, and save 300 expensive observations! c. You go to UGA and you think Dr. Pepper has a medical degree. https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 17120

10/3/21, 3:55 PM ISYyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 — Question 34 1/1pts | Who is the teacher with the least-lame sense of humor you've ever had? Correct! a. Dave Goldsman Standard Normal Table z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 | 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 | 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 | 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 03| 06179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 | 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 | 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 | 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 | 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 | 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 09 | 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 | 08413 0.8438 0.8461 0.8485 0,8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 | 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 | 0.8849 0.8869 0.8888 0.8%07 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 | 0.8032 0.9045 0.9066 0.5082 0.9099 0.9115 0.9031 0.5147 0.9162 0.5177 14 | 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 | 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 | 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 | 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 | 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 | 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 20 | 05772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 | 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 | 09861 0.9864 0.9868 0,9871 0,9875 0.9878 0.9881 0.9834 0.9887 0.9890 23 | 0.9893 0.98%6 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 24 | 09918 0.9%20 0.9922 0.9924 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 | 0.9938 0.9940 0.9%41 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 | 0.9953 0.9955 0.9956 0.9957 0.9958 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 | 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 28 | 09974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 | 0.5981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.5986 https://gatech.instructure.com/courses/187990/quizzes/279847 ?module_item_id=1796916 18/20