Week 9 Homework_ Simulation - ISYE-6644-OAN_O01_Q_ASY

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3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 1/8 Week 9 Homework Due Mar 15 at 11:59pm Points 16 Questions 16 Available Mar 8 at 8am - Mar 18 at 11:59pm Time Limit None This quiz was locked Mar 18 at 11:59pm. Attempt History Attempt Time Score LATEST Attempt 1 25 minutes 16 out of 16 Score for this quiz: 16 out of 16 Submitted Mar 8 at 7:24pm This attempt took 25 minutes.  Question 1 1 / 1 pts a. Exp( ) random variates b. Nor(0,1) random variates c. Triangular random variates d. Bern( ) random variates e. Nonhomogeneous Poisson processes Correct! f. All of the above --- and just about anything else! (f).  Question 2 1 / 1 pts Correct! a. Unif(0,1) (Lesson 7.1: Introduction to Random Variate Generation.) Unif(0,1) PRNs can be used to generate which of the following random entities? (Lesson 7.2: Inverse Transform Theorem --- Intro.) If is an Exp( ) random variable with c.d.f. , what's the distribution of the random variable ?

3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 2/8 b. Nor(0,1) c. Triangular d. Exp( ) e. None of the above Note that , where the last step follows by the Inverse Transform Theorem. Thus, the correct answer is (a).  Question 3 1 / 1 pts a. Unif(0,1) b. Nor(0,1) c. Triangular Correct! d. Exp( ) e. None of the above Since and are both Unif(0,1) (by symmetry), we have where the last step follows from Lesson 2's Inverse Transform Theorem example. Thus, the answer is (d).  Question 4 1 / 1 pts a. Uniform b. Normal c. Triangular Correct! d. Exponential e. Bernoulli (Lesson 7.2: Inverse Transform Theorem --- Intro.) If is a Unif(0,1) random variable, what's the distribution of ? (Lesson 7.2: Inverse Transform Theorem --- Intro.) Suppose that are i.i.d. Unif(0,1) random variables. Using Excel (or your favorite programming language), simulate . Draw a histogram of the 5000 numbers. What p.d.f. does the histogram look like?

3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 3/8 By the Inverse Transform Theorem, all of the 's are Exp( ). Since we have a histogram of 5000 of these, it really ought to look like an exponential p.d.f., , . Thus, the answer is (d).  Question 5 1 / 1 pts a. b. Correct! c. d. e. Set . Then , and so . Plugging in , we get . Thus, the correct answer is (c)  Question 6 1 / 1 pts Correct! a. Uniform b. Normal c. Triangular d. Exponential e. Bernoulli By the Inverse Transform Theorem, ; so the answer is (a).  Question 7 1 / 1 pts a. Unif(0,1) b. Unif(3,2) (Lesson 7.3: Inverse Transform --- Continuous Examples.) Suppose the c.d.f. of is , . Develop a generator for and demonstrate with . (Lesson 7.3: Inverse Transform --- Continuous Examples.) If is a Nor(0,1) random variate, and is the Nor(0,1) c.d.f., what is the distribution of ? (Lesson 7.3: Inverse Transform --- Continuous Examples.) If is a Unif(0,1) random variate, and is the Nor(0,1) c.d.f., what is the distribution of ?

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3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 4/8 c. Nor(0,1) d. Nor(3,2) Correct! e. Nor(3,4) By the Inverse Transform Theorem, Thus, (e) is the correct answer.  Question 8 1 / 1 pts a. b. c. Correct! d. e. None of the above Choice (a) just gives a random real number between 0 and 12. (b) gives a discrete uniform random integer from 1,2,...,12. (c) gives a continuous triangular distribution. Meanwhile, recall that we learned in class that is a 6-sided die toss. Thus, since (d) is simply the sum of two of these tosses, it is the correct answer.  Question 9 1 / 1 pts a. b. c. d. Correct! e. Both (a) and (b) f. Both (c) and (d) (Lesson 7.4: Inverse Transform --- Discrete Examples.) How would you simulate the sum of two 6-sided dice tosses? (Note that is the round-up function; and all of the 's denote PRNs.) (Lesson 7.4: Inverse Transform --- Discrete Examples.) If is Unif(0,1), how can we simulate a Geom(0.6) random variate?

3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 5/8 From the notes, we have So the answer is (e).  Question 10 1 / 1 pts a. Uniform b. Normal Correct! c. Triangular d. Exponential e. Bernoulli By the lesson notes, we know that the 5000 Xi's are all Triangular(0,1,2). Since we have a histogram of 5000 of these, it really ought to look like a triangular p.d.f. Thus, the answer is (c).  Question 11 1 / 1 pts a. Uniform b. Nor(0,1) c. Nor(5,1) d. Nor(12,2) e. Nor(41,6) Correct! f. Nor(41,18) By the lesson notes regarding the "desert island'' normal generator, we know that Thus, (Lesson 7.6: Convolution.) Suppose that and are PRNs. Let . Simulate this 5000 times, and draw a histogram of the 5000 numbers. What p.d.f. does the histogram look like? (Lesson 7.6: Convolution.) Suppose that are i.i.d. PRNs. What is the approximate distribution of ?

3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 6/8 Thus, (f) is the correct answer.  Question 12 1 / 1 pts a. Exp(1/2) b. Exp(4) c. Erlang Correct! d. Erlang e. Erlang Since and are both Unif(0,1), we have Thus, the answer is (d).  Question 13 1 / 1 pts True Correct! False Since , we have Thus, the majorizing function generally integrates to a number greater than 1, and so it cannot be a legitimate p.d.f.  Question 14 1 / 1 pts (Lesson 7.6: Convolution.) If are PRNs, what's the distribution of ? (Lesson 7.7: Acceptance-Rejection --- Intro.) In general, the majorizing function is itself a p.d.f. . (Lesson 7.9: Acceptance-Rejection --- Continuous Examples.) Suppose that is a continuous RV with p.d.f. , for . What's a good method that you can use to generate a realization of ?

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3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 7/8 a. Inversion b. Convolution c. Box-Muller Correct! d. Acceptance-Rejection e. Composition (d).  Question 15 1 / 1 pts a. 1/5 Correct! b. 5 c. 10 d. 25 e. None of the above The number of trials required is Geom( ), which has expected value . Therefore, (b) is our guy.  Question 16 1 / 1 pts a. N=0 b. N=1 c. N=2 Correct! d. N=3 e. N=4 Define . Stop as soon as . (Lesson 7.8: Acceptance-Rejection --- Proof.) Consider the constant . On average, how many iterations (trials) will the A-R algorithm require? (Lesson 7.10: Acceptance-Rejection --- Poisson Distribution.) Suppose that , , and . Use our acceptance-rejection technique from class to generate . (You may not need to use all of the uniforms.)

3/22/24, 4:05 PM Week 9 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510441?module_item_id=3598452 8/8 Take N = 3, answer is (d). Quiz Score: 16 out of 16