simulation-week-4-homework-spring-2021-simulation-isye-6644-oan

= Q studocu scarch for courses, books or documents Q « Week 4 Homework Spring 2021 Simulation ISYE 6644 OAN OO1 hw 4 University Georgia Institute of Technology B Course Simulation (ISYE 6644) B 164 documents Academic year: 2022/2023 Comments Please sign in or register to post comments. Recommended for you ~-z= Exam 2 Cheat Sheet with formulas =L = andanswers ‘@ Al Quiz ibo o [ Save @ Share 2/16/2021 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 Week 4 Homework - Spring 2021 Due Feb 19 at 11:59pm Points 14 Questions 14 Available Jan 30 at 8am - Feb 19 at 11:59pm 21 days Time Limit None Attempt History Attempt Time Score LATEST Attempt 1 4 minutes 12 out of 14 (D Correct answers will be available on Feb 20 at 12pm. Score for this o R4 @ outof 7 @& @ § o [ & Download Submitted Feb 16 at 7:45pm This attempt took 4 minutes. Incorrect Question 1 0/1pts (Lesson 3.1: Solving a Differential Equation.) Suppose that f(a:) = e2® . We know that if i is small, then fla+h)—f(=) f@) =~ =——-. Using this expression with b = 0.01, find an approximate value for f'(1) a. 1 4 (1) outefm @aa ¢ o ( &pbownload c.7.38 d. 14.93 https:/gatech instructure com/courses/165326/quizzes/213841 117 2/16/2021 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 24 (1 Jouotm @@ ¢ o [ eDownoad z+h)=J(T — Flo) ~ LT ST So using h = 0.01, we have .02 F1)~ S 1493 Thus, the answer is (d). Question 2 1/1pts (Lesson 3.1: Solving a Differential Equation.) Suppose that f(z) = e2¢.

What is the actual value of f’(1)? a. ]l b.ex~ 2.72 c.e? ~7.39 d.2¢? ~ 14.78 f'(z) = 2€**,s0 that f'(1) = 2€2, and thus the answer is (d). e.14.93 f'(z) = 2¢**,s0 that f'(1) = 2¢2, and thus the answer is (d). https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 Question 3 1/1pts (Lesson 3.1: Solving a Differential Equation.) Consider the differential equation f'(z) = (z + 1) f(x) with f(0) = 1. What is the exact formula for f(2)? c f(z) = exp{% + cc} 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 £@) _ f(z) so that F@ ff(x)dz—fzflda: Which implies z+1, In(f(z)) = & +z+C. 2 = . so that f(z) — Kez 1% where C' and K are arbitrary constants. Setting f(O) = 1 implies that K = 1, so that the exact answer is , 22 A . —+=z, i.e., choice the answer is f(a:) —e2 d. f(z) = cxp{alc2 + 2;10} (©). 217

https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 0.05 0.06 0.07 0.08 0.09 0.10 0.19 0.20 1.2287 1.2433 1.01107 1.2313 1.2461 Wow, what a good match! In any case, the answer is (c). d. £(0.20) ~ 2.49 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 By previous question, the true answer is the answer is f@)=e¥ But our job is to use Euler to come up with an iterative approximation, so here it goes. As usual, we start with fa+h) = f(@) + hf (@) = £(z) + h(@+1) f(@) = f@)1 +h(z + 1), from which we obtain the following table. z | Euler approx true f(z) 0.00 1.0000 1.0000 0.01 1.0100 1.0101 0.02 1.0202 1.0204 0.03 1.0306 1.0309 0.04 1.0412 1.0416 0.05 1.0521 1.0526 0.06 1.0631 1.0637 0.07 1.0744 1.0751 0.08 1.0859 1.0868 0.09 1.0976 1.0986 0.10 1.1096 1.01107 0.19 1.2287 1.2313 0.20 1.2433 1.2461 Wow, what a good match! In any case, the answer is (c). Question 5 1/1pts (Lesson 3.2: Monte Carlo Integration.) Suppose that we want to use Monte Carlo integration to approximate I = [',3 1 dz. If 517

Uy, Us,...,U, areiid. Unif(0,1)'s, what's I? https:/gatech instructure com/courses/165326/quizzes/213841 Discover more from: Vi 14z a good approximation ]_" for 617 « Document continues below /7 — E =0 1 2, 4 14 1o " 9 > Simulation ISYE 6644 ISYE6644 Notes T2 2020su solns All exams Exam 2 Cheat o Georgia Institute of Technology - Cheat sheet vl - Test combined - no Sheet with L) solutions thank you formulas and.. Simulation Simulation Simulation Simulation [cLRVYLITEN) 15 100% (9! 1% 100% (10) 15 100% (8) 15 100% (13 2/16/2021 Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/001 1 n 1 a o 2;‘:1 1+U; In the notation of the lesson, the general approximation we've been using is - b—a - I,= > gla+(b—a)Ui) n 1 3I-1 ¢ =—=> g1+ @E-1)U) n 4 i=1 2 n == g(1+2U;) i=1 2 & 1 n = 14+ (1+20) 1 & 1 nig1+U; so that the answer has simplified very nicely to (a). 2 n 1 b3 2ui=1 147, 1 1 C 3 2in1 1+20; 2 1 d 3 Xin Tm, 1\ 1 e Zi:l 1+30; https:/gatech instructure com/courses/165326/quizzes/213841 mr

v vauu varue wn g e a. 0.197 b.0.693 I =In(1+z)|} =In(4) — In(2) = 0.693.Thus, the answer is (b). c. 1.386 d.2.773 I=In(1+z) :{ =1n(4) — In(2) = 0.693.Thus, the answer is (b). https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 ‘Week 4 Homework - Spring 2021 Simulation - ISYE-6644-OAN/O01 Question 8 1/1pts (Lesson 3.3: Making Some 7r.) Inscribe a circle in a unit square and toss n = 1000 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 7? b. 4.0 (UGA answer) c.32 d. 3.04 The estimate #, = 4 X (proportion in circle) = 4(760/1000) = 3.04 Thus, the answer is (d). e.3.12 The estimate #, = 4 X (proportion in circle) = 4(760/1000) = 3.04Thus, the answer is (d). Question 9 1/1pts (Lesson 3.3: Making Some 7r.) Now suppose that we can somehow toss n random darts into a unit cube. Further, suppose that we've inscribed a sphere with radius 1/2 inside the cube. Letp, be the proportion of the n darts that actually fall within the sphere. Give a Monte Carlo scheme to estimate 7. https:/gatech instructure com/courses/165326/quizzes/213841 917 1017

2/16/2021 ‘Week 4 Homework - Spring 2021 Simulation - ISYE-6644-OAN/O01 a.ftn = 2p, ~ 4 ~ b.Tn = 3Pp .t = 4p, The probability that a dart falls inside the sphere is the volume of the sphere divided by the volume of the unit cube, i.e.,%m‘3 = 7r/6. Thus, for large ., we have p,, = /6, so that 7, = 6p,, should do the trick. Therefore, the answer is (d). The probability that a dart falls inside the sphere is the volume of the sphere divided by the volume of the unit cube, i.e., %m‘s = 1r/6. Thus, for large n, we have 13" ~ 7r/6, so that 7n = 6p,, should do the trick. Therefore, 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? https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 Let's make a version of our usual table. ila|n we s |b 1|5 |5 0 419 206 |13 7 4|17 1 3|89 4|13 Thus, the answer is (d). e. 19 Let's make a version of our usual table. ila|n, we s |D 1|5 |5 0 419 26 |13 7 4|17 3819 1 4|13 117

https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 Incomrect ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 By the Inverse Transform Theorem, we know that —%Z’n(l — U) ~ Exp()). Butsince 7 and 1 — [ are both Unif(0,1) (why?), we also have —3n(U) ~ Exp()). In particular, —0.5¢n(U) ~ Exp(2), so that the answer is (b). c. Exp(1/2) d. Exp(—2) e. Exp(—1/2) By the Inverse Transform Theorem, we know that —%Zn(l —U) ~ Exp()). Butsince U and 1 — U are both Unif(0,1) (why?), we also have —3n(U) ~ Exp(A). In particular, —0.56n(U) ~ Exp(2), so that the answer is (b). Question 13 0/1pts (Lesson 3.6: Simulating Random Variables.) If U; and U, are i.i.d. Unif(0,1) random variables, what is the distribution of Uy + Us ? Hints: (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 U; + U, many times and make a histogram of the results. a. Unif(0,2) https:/gatech instructure com/courses/165326/quizzes/213841 2/16/2021 ‘Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/O01 b. Normal c. Exponential d. Triangular By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d). Question 14 1/1 pts 1317 1417

(Lesson 3.7: Spreadsheet Simulation.) | 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) 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. a. $2500 https://gatech.instructure.com/courses/165326/quizzes/213841 1517 2116/2021 Week 4 Homework - Spring 2021: Simulation - ISYE-6644-OAN/001 b. $3250 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)... m = 1000000; % reps Income = 7000 + unifrnd(3000,5000,[1 m]) + unifrnd(2000,3000,[1 m]); Expenses = 13800 + unifrnd(900,1350,[1 m]) + unifrnd(600,1200,[1 m]) + unifrnd(200,600,[1 m]) + unifrnd(400,800,[1 m]); Totals = Income - Expenses; hist(Totals,100) mean(Totals) var(Totals) 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)... m = 1000000; % reps Income = 7000 + unifrnd(3000,5000,[1 m]) + unifrnd(2000,3000,[1 m]); Expenses = 13800 + unifrnd(900,1350,[1 m]) + unifrnd(600,1200,[1 m]) + unifrnd(200,600,[1 m]) + unifrnd(400,800,[1 m]); Totals = Income - Expenses; hist(Totals,100) mean(Totals) var(Totals)

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