Week 4 Homework_ Simulation - ISYE-6644-OAN_O01_Q_ASY

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Georgia Institute Of Technology *

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Apr 3, 2024

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3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 1/12 Week 4 Homework Due Feb 9 at 11:59pm Points 13 Questions 13 Available Feb 2 at 8am - Feb 12 at 11:59pm Time Limit None This quiz was locked Feb 12 at 11:59pm. Attempt History Attempt Time Score LATEST Attempt 1 36 minutes 13 out of 13 Score for this quiz: 13 out of 13 Submitted Feb 5 at 8:34pm This attempt took 36 minutes.  Question 1 1 / 1 pts a. 1 b. 2.72 c. 7.38 Correct! d. 14.93 We have So using h = 0.01, we have Thus, the answer is (d). We have (Lesson 3.1: Solving a Differential Equation.) Suppose that . We know that if is small, then Using this expression with , find an approximate value for .

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 2/12 So using h = 0.01, we have Thus, the answer is (d).  Question 2 1 / 1 pts a. b. c. Correct! d. and thus the answer is (d). e. and thus the answer is (d).  Question 3 1 / 1 pts a. b. Correct! c. This takes a little work. The good news is that you can actually get the true answer using the technique of separation of variables. We have so that Which implies (Lesson 3.1: Solving a Differential Equation.) Suppose that . What is the actual value of ? (Lesson 3.1: Solving a Differential Equation.) Consider the differential equation with . What is the exact formula for ?

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 3/12 , so that , where and are arbitrary constants. Setting implies that , so that the exact answer is , the answer is , i.e., choice (c). d. This takes a little work. The good news is that you can actually get the true answer using the technique of separation of variables. We have so that Which implies , so that , where and are arbitrary constants. Setting implies that , so that the exact answer is , the answer is , i.e., choice (c).  Question 4 1 / 1 pts a. b. Correct! c. By previous question, the true answer is the answer is . But our job is to use Euler to come up with an iterative approximation, so here it goes. As usual, we start with from which we obtain the following table. (Lesson 3.1: Solving Differential Equations.) Consider the differential equation with . Solve for using Euler's approximation method with increment for .

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3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 4/12 Wow, what a good match! In any case, the answer is (c). d. By previous question, the true answer is the answer is . But our job is to use Euler to come up with an iterative approximation, so here it goes. As usual, we start with from which we obtain the following table.

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 5/12 Wow, what a good match! In any case, the answer is (c).  Question 5 1 / 1 pts Correct! a. In the notation of the lesson, the general approximation we've been using is (Lesson 3.2: Monte Carlo Integration.) Suppose that we want to use Monte Carlo integration to approximate . If are i.i.d. Unif(0,1)'s, what's a good approximation for ?

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 6/12 so that the answer has simplified very nicely to (a). b. c. d. e. In the notation of the lesson, the general approximation we've been using is so that the answer has simplified very nicely to (a).  Question 6 1 / 1 pts

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3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 7/12 a. 0 b. 0.2 c. 0.321 Correct! d. 0.679 , so the answer is (d). e. 0.8 , so the answer is (d).  Question 7 1 / 1 pts a. 0.197 Correct! b. 0.693 Thus, the answer is (b). c. 1.386 d. 2.773 Thus, the answer is (b).  Question 8 1 / 1 pts a. (Lesson 3.2: Monte Carlo Integration.) Again suppose that we want to use Monte Carlo integration to approximate . You may have recently discovered that the MC estimator is of the form Estimate the integral by calculating with the following 4 uniforms: (Lesson 3.2: Monte Carlo Integration.) Yet again suppose that we want to use Monte Carlo integration to approximate . What is the exact value of ? (Lesson 3.3: Making Some .) Inscribe a circle in a unit square and toss random darts at the square. Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for ?

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 8/12 b. 4.0 (UGA answer) c. 3.2 Correct! d. 3.04 The estimate Thus, the answer is (d). e. 3.12 The estimate Thus, the answer is (d).  Question 9 1 / 1 pts a. 3 b. 9 c. 13 Correct! d. 17 Let's make a version of our usual table. Thus, the answer is (d). e. 19 Let's make a version of our usual table. Thus, the answer is (d).  Question 10 1 / 1 pts (Lesson 3.4: Single-Server Queue.) Consider a single-server Q with LIFO ( last -in-first-out) services. Suppose that three customers show up at times 5, 6, and 8, and that they all have service times of 4. When does customer 2 leave the system?

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 9/12 a. b. Correct! c. 44 Thus, the answer is (c). d. 45 e. 70 Thus, the answer is (c).  Question 11 1 / 1 pts a. Who knows? Correct! b. Exp(2) By the Inverse Transform Theorem, we know that . But since and are both Unif(0,1) (why?), we also have In particular, so that the answer is (b). (Lesson 3.5: Inventory Model.) Consider our numerical example from the lesson. What would the third day's total profits have been if we had used a (4,10) policy instead of a (3,10)? (Lesson 3.6: Simulating Random Variables.) If is a Unif(0,1) random number, what is the distribution of ?

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3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 10/12 c. Exp(1/2) d. Exp e. Exp By the Inverse Transform Theorem, we know that . But since and are both Unif(0,1) (why?), we also have In particular, so that the answer is (b).  Question 12 1 / 1 pts a. Unif(0,2) b. Normal c. Exponential Correct! d. Triangular By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d). By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d).  Question 13 1 / 1 pts (Lesson 3.6: Simulating Random Variables.) If and are i.i.d. Unif(0,1) random variables, what is the distribution of ? Hints: (i) I may have mentioned this in class at some point; (ii) You may be able to reason this out by looking at the distribution of the sum of two dice tosses; or (iii) You can use something like Excel to simulate many times and make a histogram of the results. (Lesson 3.7: Spreadsheet Simulation.) I stole this problem from the Banks, Carson, Nelson and Nicol text (5th edition). Expenses for Joey's college attendance next year are as follows (in $): Tuition = 8400 Dormitory = 5400 Meals Unif(900,1350) Entertainment Unif(600,1200) Transportation Unif(200,600) Books Unif(400,800)

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 11/12 a. $2500 b. $3250 Correct! c. $3325 An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c). If you don't believe me, here's some Matlab code (if you happen to have Matlab)... d. $3450 e. $4000 An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c). If you don't believe me, here's some Matlab code (if you happen to have Matlab)... Here are the income streams the student has for next year: Scholarship = 3000 Parents = 4000 Waiting Tables Unif(3000,5000) Library Job Unif(2000,3000) Use Monte Carlo simulation to estimate the expected value of the loan that will be needed to enable Joey to go to college next year.

3/22/24, 3:50 PM Week 4 Homework: Simulation - ISYE-6644-OAN/O01/Q/ASY https://gatech.instructure.com/courses/360616/quizzes/510307?module_item_id=3598400 12/12 Quiz Score: 13 out of 13

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Week 4 Homework_ Simulation - ISYE-6644-OAN_O01_Q_ASY

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