ISYE 6644 - Summer 2024 - Final Exam Questions and Answers with complete solution

ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 ISyE 6644 – Summer 2024 - Final Exam Due Aug 5 at 11:59pm Points 100 Questions 34 Available Jul 28 at 8am - Aug 13 at 11:59pm 17 days Time Limit 120 Minutes Instructions This test is 120 minutes. You're allowed three cheat sheets (6 sides total). You may use a calculator. You can find Normal, t and Chi-square tables on the last page of this exam (although you may not need them all - or at all!). You will not need Kolmogorov-Smirnov or ranking-andselection tables. Note that you are not allowed to use Arena (or any other software), even though I'm asking questions about it. If you run into a technical issue during your exam, please contact Canvas: Canvas: Call or use the chat chattype=student ( HonorLock: Use the chat function in the HonorLock window. BEFORE YOU TAKE THIS QUIZ You will need to complete verification. © 2021 Honorlock Inc. Support ( ( s eoxfaSmervriceeq(hutitrpes:s//hGoo noorlgolcek.cCohmr/loegmale/tearn mds) the This quiz is no longer available as the course has been concluded.ISyE 6644 – Summer 2024 - 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. Correct! Correct! Question 1 3 / 3 pts SupposeX andY have joint p.d.f. f(x,y) = 8xy for all 0 < y < x < 1. Find E[5X − 1]. 4 3 a. 1/5 b. 2/3 c. 4/5 d. 5/6 e. 1 Question 2 3 / 3 pts YES or NO? Consider again the joint p.d.f. from the previous question, f(x,y) = 8xy for 0 < y < x < 1. Are X and Y dependent random variables? a. Yes b. NoISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! orrect Answ ou Answered Question 3 3 / 3 pts IfX andY are i.i.d. Nor(1,2) random variables, find Var(X − Y + 2). a. 0 b. 0.5 c. 1 d. 2 e. 4 Question 4 0 / 3 pts IfZ1,Z2,⋯,Z2n are i.i.d. Nor(0,1), what is the distribution of ∑2n Z2 ? i=1 i a. Nor(0, 2n) b. Pois(2n) er c. χ2(2n) d. Both (b) and (c) e. All of the above.ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 7 3 / 3 pts Question 5 3 / 3 pts SupposeU1 and U2 are i.i.d. Unif(0,1). What is the distribution of 5U1 − 5U2 ? a. Unif(−5, 5) b. Unif(0, 10) c. Tria(−10, 0, 10) d. Tria(−5, 0, 5) e. Tria(0, 5, 10) Question 6 3 / 3 pts SupposeU1 and U2 are i.i.d. Unif(0,1), and let X = √−2ln(U1)cos(2πU2). Find Pr(X < −1). a. 0 b. 0.1587 c. 0.50 d. 0.8413 e. 1.00ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 9 3 / 3 pts IfU1 and U2 are i.i.d. Unif(0,1), what is the distribution of −0.5 ln[(1 − U1)4U 4] ? 2 a. Exp(0.5) b. Erlang2(2) c. Erlang2(−2) d. Erlang2(0.5) e. Erlang2(−0.5) Question 8 3 / 3 pts Suppose we are given a choice between two estimators, T1 and T2, and are told that the relative efficiency of T1 to T2 is 0.8. That is, the Mean Square Error (MSE) ratio between the two estimators is MSE(T2) = 1.8. MSE(T1) Which estimator would you choose, and why? a. Choose T1, it has a lower MSE. b. Choose T1, it has a higher MSE. c. Choose T2, it has a lower MSE. d. Choose T2, it has a higher MSE. e. Either estimator would be an equally acceptible choice.ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! ou Answered orrect Answ Question 11 0 / 3 pts Question 10 0 / 3 pts Suppose that we take three i.i.d. observationsX1 = 0.6,X2 = 2.4, and X3 = 3, from X ∼ Exp(λ). Using the maximum likelihood estimate for λ , find the MLE of Pr(X ≤ 2). a. e− 1 2 b. 1 − e− 1 2 c. e−1 er d. 1 − e−1 e. 1 − e−2ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Consider a nonhomogeneous Poisson arrival process with rate function λ(t) = t/2 fort ≥ 0. Find the probability that there will be more than two arrivals before time t = 2. a. 0.0613 orrect Answer b. 0.0803 c. 0.1839 ou Answered d. 0.5518 e. 0.9197 Question 12 3 / 3 pts Consider a 2 × 2 covariance matrix Σ =(3 2). Calculate the lower- 2 5 triangular matrix C such that CC′ = Σ, and tell me the value of the entry c21 . a. 0 b. 3 c. 11/3 d. √11/3 e. 2/√3ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! ou Answered orrect Answ Question 13 3 / 3 pts 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. W(10) − W(7) is independent of W(3) − W(2) d. A Brownian Bridge, B(t), is a conditioned BM such that W(0) = W(1) = 0 e. Cov(W(1), W(3)) = 1 Question 14 0 / 3 pts Let W(t) denote a Brownian motion process at time t . Let A = ∫ 1 W(t) dt represent the area under the process from time t = 0 0 tot = 1. Find the probability that A ≤ 1. a. 0.6293 b. 0.7257 c. 0.8413 er d. 0.9584 e. 0.9817ISyE 6644 – Summer 2024 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 17 0 / 3 pts TRUE or FALSE? In ARENA, a DECIDE module can only make decisions on where to go next based on chance/probabilities. Question 15 3 / 3 pts Suppose X1, … ,X100 are i.i.d. from an Exp(1) distribution. Use the Central Limit Theorem to find an approximate distribution of the sample mean X100 and then calculate Pr(X100 ≤ 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. True False10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 ou Answered orrect Answ Correct! Correct! Question 20 3 / 3 pts Question 18 3 / 3 pts TRUE or FALSE ? In ARENA, entities must be CREATE'd to get into the model, and DISPOSE'd to leave. (Don't overthink this problem!) True False Question 19 3 / 3 pts TRUE or FALSE? In ARENA, queues cannot be defined manually; they can only be (automatically) created by certain modules (e.g., PROCESS or SEIZE modules). True False10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 22 3 / 3 pts Suppose we're conducting a χ2 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 Question 21 3 / 3 pts Suppose we're conducting a χ2 goodness-of-fit test to determine whether or not 1500 i.i.d. observations are from a shifted Gamma(α,β,c) distribution, where α, β, and the shift parameter c must all be estimated. If we divide the observations into k = 7 equal-probability intervals, how many degrees of freedom will our test have? a. 3 b. 6 c. 7 d. 11 e. 149710/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! orrect Answ ou Answered Correct! Question 23 0 / 3 pts TRUE or FALSE? Simulation output (e.g., consecutive waiting times) is almost never i.i.d., nor normal. er True False Question 24 3 / 3 pts TRUE or FALSE? Steady-state analysis should only be performed once simulation output initialization bias effects are dealt with (or at least considered). True False10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! orrect Answ ou Answered Question 27 3 / 3 pts Question 25 3 / 3 pts Consider a particular data set of 30000 stationary waiting times obtained from a large queueing system. Suppose your goal is to get a confidence interval for the unknown mean. Would you rather use (a) 30 batches of 1000 observations or (b) 3000 batches of 10 observations each? a. 30 batches of 1000 observations each b. 3000 batches of 10 observations each Question 26 0 / 3 pts Consider the following 12 observations arising from a simulation: 169 166 Use the method of batch means to calculate a two-sided 95% confidence interval for the mean μ. In particular, use two batches of size six. a. [54.613, 296.573] er b. [139.735, 203.265] c. [163.634, 179.366] d. 171.5 ± 32.186 e. 171.5 ± 4510/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 29 3 / 3 pts Question 28 3 / 3 pts Which variance reduction technique is most closely associated with a paired-t confidence interval for the mean? a. Antithetic Random Numbers b. Common Random Numbers c. Control Variates d. Both (a) and (b) e. None of the above10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 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. Question 30 0 / 3 pts Replication System 1 System 2 1 10 25 2 20 40 3 25 30 4 35 20 Correct! Replication System 1 System 210/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 ou Answered orrect Answer 1 10 20 2 20 25 3 25 30 4 35 40 orrect Answ ou Answered Question 32 3 / 3 pts Question 31 0 / 3 pts What would you use a ranking and selection method for? a. Conducting a goodness-of-fit test. b. Finding a confidence interval for the mean. er c. Finding the best of a number of competing systems. d. Estimating the power of a hypothesis test. e. Determining a good truncation point for a steady-state simulation.10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Correct! Question 33 3 / 3 pts 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 δ∗, 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.10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Correct! Question 34 1 / 1 pts Who is the teacher with the least-lame sense of humor you've ever had? a. Dave Goldsman10/3/21, 3:55 PM ISyE 6644 - Summer 2021 - Final Exam : Simulation - ISYE-6644-A/OAN/O01 Quiz Score: 73 out of 100