Hi. I need help with a Risk and Decision Analysis course where we work a lot with probability. Now I am struggling with Maximum Likelihood Estimation problems. I have an assignment due next Monday 10th and would really appreciate if you can help me with it. It also involves working with R studio. Please let me know. Thank you!
Sheet1 SampleTime (Weeks)Time in-between accidents (weeks) 114 23319 3429 46018 59535 616570 71738 818916 91978 1021013 1123424 1225521 1326712 1428013 1529818 163068 173159 1834227 1936220 2043775 2146427 224739 2355784 2458528 2559611 288.958333333324.25 Sheet2 11 0.870.7569 24 1.281.6384 0.740.5476 1.011.0201 0.920.8464 0.80.64 1.051.1025 0.160.0256 1.662.7556 0.80.64 14.9731 29.9462 0.0333932185 0.1827381145 Homework 2 CIVL 519: Risk and Decision Analysis for Infrastructure Management Term 1 – 2022/2023 Due Date: 11:59 PM, October 10, 2022 Instructions for the submission can be found on Canvas Problem No. Score Maximum Possible Points 1 2.25 2 2 3 1.5 4 2.25 Total: 8 CIVL 519 Homework 1 Term 1 2022 1/4 Problem 1: TransLink has reached out to you to conduct a safety study for a particular intersection. You can download their (hypothetical) data from our Canvas site. So far, a total of 25 accidents have occurred at this intersection. For each accident, they have reported to you the week in which the accident occurred relative to the date that they began tracking these data. For example, the time between Accidents 1 and 2 was 19 weeks (i.e., 33 Weeks – 14 Weeks). You would like to construct a simulation model of traffic accidents at this intersection moving into the future both with and without changes to its design. To create this simulation model, you would like to be able to sample the inter-arrival time of accidents as following the below distribution: You would like to begin your analysis by first analyzing your data to get a sense of some “reasonable” parameter estimates. Question A: Derive the maximum likelihood estimator for θ1 and θ2 for any arbitrary set of data. Show all your steps to derive it. Question B: What is the maximum likelihood estimate for θ1 and θ2 for your data? Question C: Holding θ2 constant, plot the log-likelihood estimate of θ1 for your data over a range of values that are both above and below your maximum likelihood estimate. Do your results make sense? Provide a 1-2 sentence comment. Question D: Do you have (or not have) any hesitations about using this type of probability density function for your model? Provide a 1 paragraph comment. CIVL 519 Homework 1 Term 1 2022 2/4 Problem 2: You are an equipment operator. Suppose that you believe that the lifetime, x, for a typical piece of equipment for your operations follows the below probability density function: Question A: For a given piece of equipment, what is the reliability, R(x), that it will still be operating at time x. View R(x) as the probability that a piece of equipment will still be operating at time x. Question B: What is the maximum likelihood estimator of R(x)? Suppose that you have collected data on five pieces of equipment, whose lifetime were {5 years; 8 years; 10 years; 4 years; 8 years}. Question C: Based on your maximum likelihood estimate, plot R(x). Question D: Based on your maximum likelihood estimate, simulate x over 1,000 Monte Carlo simulations. Overlay on your plot from Question C your estimate of R(x) from the Monte Carlo simulations. Is your estimate of R(x) via Monte Carlo simulation similar to your plot from Question C? CIVL 519 Homework 1 Term 1 2022 3/4 Problem 3: As a system modeler at BC Hydro, you are charged with forecasting future precipitation in the region. Assume that the amount of precipitation, x, in a given year follows the below distribution: Question A: Derive the maximum likelihood estimator for θ. Suppose that you have accessed the following precipitation data (in meters) for the last 12 years: {1.00, 0.87, 2.00, 1.28, 0.74, 1.01, 0.92, 0.80, 1.05, 0.16, 1.66, 0.80}. Question B: Based on Question A, what is your best estimate of θ? CIVL 519 Homework 1 Term 1 2022 4/4 Problem 4: For this problem, we will actually rely on real data! Suppose that you are a cost estimator for the BC MOTI. To maintain its 80,000+ lane-kilometers of paved surfaces in a good state-of-repair, the agency must spend significant sums of money each year on pavement overlays. To improve its budgeting process, you have been charged with improving the MOTI’s estimate of the unit-cost of overlay treatments. Based on the file available to you on Canvas, I have provided you: • Column A: A project identifier • Column B: The year the project took place • Column C: The quantity of asphalt (i.e., number of tons) for that project • Column D: The unit-cost of asphalt (i.e., $/ton) for that project Question A: Fit the unit-cost data to a normal, log-normal, Weibull and Gamma distribution. Report the log-likelihood estimate for each distribution. What is your preferred distribution based on the log-likelihood estimate (1 sentence)? Question B: Generate a P-P or Q-Q plot for each distribution. Has your preferred distribution changed? Provide a 1-2 sentence comment. Suppose that a typical overlay project is 1-mile long. There will be two lanes, each 12 feet wide. The thickness of an asphalt overlay is typically 2-inches. Furthermore, the density of asphalt is approximately 150 pounds per cubic foot. Question C: Suppose that you believe that the unit-cost of asphalt follows your fitted normal distribution. Simulate and plot your CDF of the total cost (i.e., $/ton x tons) for a typical pavement project. Do your results make sense? Do you see any modeling issues? Provide a 1-paragraph discussion. Question D: Suppose that you believe that the unit-cost of asphalt follows your fitted lognormal distribution. Plot your CDF of the total cost (i.e., $/ton x tons) for a typical pavement project. Do your results make sense? Do you see any modeling issues? How does it compare to your answer for Question C? Provide a 1-paragraph discussion. Question E: Create two scatter plots with your data: (1) unit-cost vs. time; and (2) unit-cost vs. quantity. Based on what you have observed, do you believe that unit-cost is independent of either of these factors? Would your findings suggest that you revisit your modeling approach? 1-2 paragraph discussion.