Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
OR 664 / SYST 664 / CSI 674: Final Exam
Each question is worth ten points, for a total of 100 points for the final exam. Show all your
reasoning. You will receive some credit for making an honest attempt and more credit if I can
tell you were thinking along the right lines. Write up your solutions in a pdf file and submit to
Gradescope. Your writeup for each problem should be self-contained and may include sections
of code. You may submit other documents (R file, spreadsheet) separately to Blackboard for
reference, but your solution must be understandable on its own. You are bound by the GMU
honor code to work by yourself on the exam. I will be available by phone or email to answer
clarification questions.
1. A sensor is designed to detect a pollutant in soil samples. The manufacturer claims the
sensor has 85% accuracy. Based on this claim, the following prior distribution has been
specified for the true positive and false positive rates:
• True positive rate (sensitivity). The probability of correctly detecting the pollutant if it
is present has a beta distribution with shape parameters 4.25 and 0.75.
• False positive rate (1 – specificity). The probability that the test will erroneously
report the pollutant if it is not present is independent of the sensitivity and has a beta
distribution with shape parameters 0.75 and 4.25.
A test was performed on 35 soil samples known to contain the pollutant and 40 soil samples
known not to contain the pollutant. The sensor correctly detected 26 out of the 35 samples
containing the pollutant, and incorrectly reported the pollutant in 7 of the samples not
containing the pollutant. What is the joint posterior distribution for the true positive and false
positive rate of the sensor? Find 95% posterior credible intervals for the true and false
positive rates.
2. A scientist is studying the effect of soil conditions on the height of a species of plant. Height
data in centimeters was collected on four different plots of land with different soil conditions.
Plot A Plot B Plot C Plot D
21.1 19.2 27.0 20.1
30.5 35.7 23.6 42.4
19.3 16.7 19.3 30.5
14.8 18.6 8.8 32.2
17.5 23.8 21.9 28.3
12.0 24.6 10.8 23.8
23.4 25.9 12.4 25.8
34.1 10.1 10.6 15.5
16.7 21.7 20.2 19.8
11.4 17.7 7.9 23.3
14.3 18.6 16.0 23.0
25.0 37.4 20.5 33.3
13.9 27.1 1.1 16.6
21.9 29.5 16.6 28.5
21.5 32.7 26.9 18.8
OR 664 / SYST 664 / CSI 674 Final Exam Page 2 Spring 2021
Assume the observations from each plot are normally distributed with unknown plot-specific
means Qi and precisions Ri for i=1, …, 4. Assume the parameters (Qi, Ri) are independent
draws from a normal-gamma(µ, k, a, b) distribution. Find empirical Bayes estimates for the
hyperparameters µ, k, a, and b as follows:
• Estimate the center µ as the grand mean of all the observations.
• Estimate the shape and scale as follows. Estimate the sample precisions "!, "", "#, "$
by calculating the sample variances and taking their inverses. Estimate the mean
of the Gamma distribution as the average of "!, "", "#, "$. Estimate the variance " as the sample variance of "!, "", "#, "$. Then solve for and .