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Georgia Institute Of TechnologyISYE 6644Week 11 Homework_ Simulation - ISYE-6644-OAN

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12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 1/8 Week 11 Homework Due Nov 10 at 11:59pm Points 9 Questions 10 Available after N... ov 1 at 8am Time Limit None Attempt History Attempt Time Score LATEST Attempt 1 110 minutes 6 out of 9 Score for this quiz: 6 out of 9 Submitted Nov 3 at 1:12pm This attempt took 110 minutes. Question 1 0 / 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, Y You Answered ou Answered d. If , then ; otherwise, orrect Answer orrect Answer e. If , then ; otherwise, 2/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 2/8 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 2 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. 1.96 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). 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 3/8 Question 3 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. 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). 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 4/8 Question 4 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. 1 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 5 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 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 5/8 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 Y You Answered ou Answered b. 0.64 orrect Answer orrect Answer c. 0.72 d. 0.8 e. 1 I'll actually give the analytical solution. This is answer (c). Question 6 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 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 6/8 c. 0.5 d. 0.805 e. 1 I'll give the analytical solution. First of all, the number of arrivals in that time interval is Thus, This is answer (b). Question 7 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. Y You Answered ou Answered a. 0 b. 0.7 c. 0.49 orrect Answer orrect Answer d. e. 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 7/8 By class notes, the analytical answer is , so that (d) is correct. Question 8 1 / 1 pts (Lesson 7.19: Brownian Motion.) Let denote a Brownian motion process at time . Calculate . a. 0 b. 2 Correct! Correct! c. 3 d. 5 e. 8 We have , so that the answer is (c). Question 9 1 / 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... a. -1/2 b. 0 12/7/2019 Week 11 Homework: Simulation - ISYE-6644-OAN https://gatech.instructure.com/courses/75320/quizzes/69201 8/8 Correct! Correct! c. 1/12 d. 1/2 e. 1 Question 10 0 / 0 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 Aug 16, 2019, IBM is currently selling for about $134 per share. Suppose I'm interested in guaranteeing that I can buy a share of IBM for at most $145 on Nov. 15, 2019. Look up (maybe using something like FaceTube on the internets) the corresponding stock option price. Did you look up the answer? (Just write "YES".) As I was writing this solution sheet, the option price was $2.02 --- but this is obviously subject to change depending on how the market does. Correct! Correct! a. Yes b. No Quiz Score: 6 out of 9 [Show More]

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