Sheet1
Sample Time (Weeks) Time in-between accidents (weeks)
1 14
2 33 19
3 42 9
4 60 18
5 95 35
6 165 70
7 173 8
8 189 16
9 197 8
10 210 13
11 234 24
12 255 21
13 267 12
14 280 13
15 298 18
16 306 8
17 315 9
18 342 27
19 362 20
20 437 75
21 464 27
22 473 9
23 557 84
24 585 28
25 596 11
XXXXXXXXXX 24.25
Sheet2
1 1
0.87 0.7569
2 4
1.28 1.6384
0.74 0.5476
1.01 1.0201
0.92 0.8464
0.8 0.64
1.05 1.1025
0.16 0.0256
1.66 2.7556
0.8 0.64
14.9731
29.9462
XXXXXXXXXX
XXXXXXXXXX
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 occu
ed at this intersection. For each accident, they have reported to you the week in which
the accident occu
ed relative to the date that they began tracking these data. For example, the time
etween 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-a
ival 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 a
itrary 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
elow 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 prefe
ed distribution based
on the log-likelihood estimate (1 sentence)?
Question B: Generate a P-P or Q-Q plot for each distribution. Has your prefe
ed 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.