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?