[SOLVED] Isye 6420 spring 2025 homework 3

Q1
Marietta Traffic Authority is concerned about the repeated accidents at the intersection of Canton and
Piedmont Roads. Bayes-inclined city-engineer would like to estimate the accident rate, even better, find a
credible set.
A well-known model for modeling the number of road accidents in a particular location/time window is
the Poisson distribution. Assume that X represents the number of accidents in a 3 month period at the
intersection of Canton and Piedmont Roads.
Assume that [X | θ] ∼ Poi(θ). Nothing is known a priori about θ, so it is reasonable to assume the Jeffreys
prior
π(θ) = 1
√
θ
1(0 < θ < ∞)
In the four most recent three-month periods the following realizations for X are observed: 1, 2, 0, and 2 .
(a) Compare the Bayes estimator for θ with the MLE (For Poisson, recall, ˆθMLE = X¯ ).
(b) Compute the 95% equitailed credible set.
(c) Compute (numerically) the 95% HPD credible set. Use an optimization method, not a sampling method.
(d) Numerically find the mode of the posterior, that is, MAP estimator of θ. Make sure it matches the result
of the known equation for the posterior mode.
(e) If you test the hypotheses
H0 : θ ≥ 1 vs H1 : θ < 1
based on the posterior, which hypothesis will be favored?
(f) Derive the posterior predictive distribution. Based on this, how many accidents do you predict for the
next year?
Q2
Waiting time. The waiting time for a bus at a given corner at a certain time of day is known to have a
U(0, θ) distribution. It is desired to test H0 : 0 ≤ θ ≤ 15 versus H1 : θ > 15. From other similar routes, it
is known that θ has a Pareto (5, 3) distribution. If waiting times of 10, 8, 10, 5, and 14 are observed at the
given corner, calculate the posterior odds ratio and the Bayes factor. Which hypothesis would you favor?
Note: the density of a Pareto distribution with parameters (c, α) is given by
αcα
θ
α+1 1(θ > c)
Q3
The Maxwell distribution with parameter α > 0, has a probability density function for x > 0 given by
p(x | α) = r
2
π
α
3/2x
2
exp 
−
1
2
αx2

(a) Find the Jeffreys prior for α.
(b) Find a transformation of this parameter in which the corresponding prior is uniform. (Hint: See Section 5.7 of Engineering Biostatistics).
(c) Find the posterior distribution for n independent and identically distributed datapoints x1, . . . , xn from
the Maxwell distribution, assuming the Jeffreys prior on α from part (a).