Consider the arrival times data of Problem 1 and again take α = 3 but change the prior to be uniform, namely,

so that

Find the Bayes estimator given by (16.15) and compare your answer to that of Problem 1. What is the posterior probability that the bus will be more than 10 min late? Is the unknown prior reasonable for a situation where a priori the bus has a tendency to be late?

Problem 1

The morning arrival times for 10 days of a school bus at the Tipperary street pickup stop are 8:05, 8:06, 8:09, 8:12, 8:03, 8:15, 8:11, 8:16, 8:00, and 8:14. The scheduled pickup time is 8:05 but past experience has led the students to believe there is a tendency for the bus to arrive late rather than be on time or early. For convenience we relabel so that 8:00 is represented by 0, 8:05 by 5, and so forth, hence we can consider the possible arrival interval to be [0, 20]. To utilize the students' experience, we choose the prior where the prior density μ' = f₀ is

This choice makes arrivals a priori three times more likely to be in the interval [10, 20] than in the interval [0, 10]. The corresponding prior distribution μ is

For the choice α = 3, find the Bayes estimator given by (16.15). What is the posterior probability that the bus will be more than 10 min late?