Georgia Institute Of TechnologyISYE 6644Homework 8 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science

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8/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 1/11... Week 8 Homework - Summer 2020 Due Jul 10 at 11:59pm Points 11 Questions 11 Available Jul 3 at 8am - Jul 10 at 11:59pm 8 days Time Limit None This quiz was locked Jul 10 at 11:59pm. Attempt History Attempt Time Score LATEST Attempt 1 17 minutes 7 out of 11 Score for this quiz: 7 out of 11 Submitted Jul 10 at 11:22pm This attempt took 17 minutes. Question 1 1 / 1 pts (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.) a. N=0 b. N=1 c. N=2 Correct! Correct! d. N=3 e. N=48/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 2/11 Define . We'll stop as soon as . Let's make the following convenient table. So we take N = 3, and the answer is (d). Question 2 1 / 1 pts (Lesson 7.11: Composition.) BONUS: It's Raining Cats and Dogs is a pet store with 60% cats and 40% dogs. The weights of cats are Nor(12,4), and the weights of dogs are Nor(30,25). How would we use composition to simulate the weight of a random pet from the store? (Let denote the standard normal c.d.f., and let 's denote PRN's.) a. b. c. If , then ; otherwise, d. If , then ; otherwise, Correct! Correct!8/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 3/11 e. If , then ; otherwise, By inverse transform, the weight of a cat all by itself is Similarly, the weight of a dog all by itself is . These facts eliminate choices (a), (c), and (e). In addition, (a) and (b) are some kind of mutant cat-dog, both of which sort of combine 0.6 of a cat with 0.4 of a dog; so those are wrong. What we really want is to take with probability 0.6, and otherwise . This is choice (d)! Question 3 1 / 1 pts (Lesson 7.12: The Box-Muller Method.) Suppose and are i.i.d. Unif(0,1) with and . Use the "cosine" version of BoxMuller to generate a single Nor(-1,4) random variate. Don't forget to use radians instead of degrees! a. -0.326 b. 0 Correct! Correct! c. 0.326 d. 0.663 e. 1.968/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 4/11 Box-Muller immediately gives the following Nor(0,1) random variate: To obtain the realization of the Nor($-$1,4), we simply apply the transform which is choice (c). Question 4 1 / 1 pts (Lesson 7.13: Generating Order Statistics.) Consider i.i.d. Exp( ) random variables , and let . How can we generate using just one PRN? a. b. Correct! Correct! c. d. e.8/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 5/11 Let F denote the Exp( ) c.d.f. Recall that the inverse is a fact that we'll use in a minute. Meanwhile, as also explained in class, the c.d.f.\ of $Y$ is Now, by inverse transform, , and so . This implies that , and thus where the last equality follows from . This is choice (c). Question 5 1 / 1 pts (Lesson 7.14: Multivariate Normal Distribution.) Suppose I have a matrix . Find the lower triangular matrix such that and tell me what the entry is. a. -1 Correct! Correct! b. -0.7071 c. 0 d. 0.7071 e. 18/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 6/11 From our class notes on multivariate normal random variate generation, we know that the Cholesky matrix we need is Therefore, the correct answer is (b). Question 6 0 / 1 pts (Lesson 7.15: Baby Stochastic Processes.) BONUS: Consider a Markov chain in which if it rains on day ; and otherwise, . Denote the day-to-day transition probabilities by Suppose that the probability state transition matrix is Suppose that it rains on Monday, e.g., . Use simulation to find the probability that it rains on Wednesday, e.g., estimate . [You may have to simulate the process a bunch of times in order to estimate this probability.] a. 0 ou Answered ou Answered b. 0.64 orrect Answer orrect Answer c. 0.72 d. 0.8 e. 18/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 7/11 I'll actually give the analytical solution. This is answer (c). Question 7 1 / 1 pts (Lesson 7.16: Nonhomogeneous Poisson Processes.) Suppose that the arrival pattern to a parking lot over a certain time period is an NHPP with . Use simulation to find the probability that there will be exactly 3 arrivals between times and . a. 0 Correct! Correct! b. 0.195 c. 0.5 d. 0.805 e. 18/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 8/11 I'll give the analytical solution. First of all, the number of arrivals in that time interval is Thus, This is answer (b). Question 8 0 / 1 pts (Lesson 7.17: Time Series Generation. ) BONUS: Suppose that and consider the time series , , where the 's are i.i.d. Nor . (The funny variance of guarantees that for all ). Use simulation to find . Hint: Simulate many times. For each run of the simulation, save the pair . Then use those pairs to estimate the covariance. a. 0 ou Answered ou Answered b. 0.7 c. 0.49 orrect Answer orrect Answer d. e.8/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 9/11 By class notes, the analytical answer is , so that (d) is correct. Question 9 0 / 1 pts (Lesson 7.19: Brownian Motion.) Let denote a Brownian motion process at time . Calculate . a. 0 ou Answered ou Answered b. 2 orrect Answer orrect Answer c. 3 d. 5 e. 8 We have , so that the answer is (c). Question 10 0 / 1 pts (Lesson 7.19: Brownian Motion.) Let denote a Brownian motion process at time and define a Brownian bridge by for . Find the variance of the area under a bridge, i.e., . I'm a nice guy, so I'll get you started...8/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 10/11 ou Answered ou Answered a. -1/2 b. 0 orrect Answer orrect Answer c. 1/12 d. 1/2 e. 18/3/2020 Week 8 Homework - Summer 2020: Simulation and Modeling for Engineering and Science - ISYE-6644-OAN/O01 https://gatech.instructure.com/courses/128642/quizzes/142113?module_item_id=833706 11/11 Question 11 1 / 1 pts (Lesson 7.19: Brownian Motion.) As we discussed in class, you can use Brownian motion to estimate option prices for stocks. I'm not going to have you simulate that, but I'm going to give you a quick look-up assignment. As I write this on May 10, 2020, IBM is currently selling for about $122.99 per share. Suppose I'm interested in guaranteeing that I can buy a share of IBM for at most $145 on Sept. 20, 2020. Look up (maybe using something like FaceTube on the internets) the corresponding stock option price. [You don't have to write down an answer for this problem, but I'd like you to do the look-up anyway.] As I was writing this solution sheet, the option price was $2.72 --- but this is obviously subject to change depending on how the market does. Correct! Correct! a. I swear I looked it up :) b. I was lazy and didn't look it up :( Quiz Score: 7 out of 11 [Show More]

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