Alice and Bob arrange to meet for lunch on a certain day at noon. However, neither is known for punctuality. Each is willing to wait up to 15 minutes for the other to show up.
(a) They both arrive independently at uniformly distributed times between noon and 1 pm on that day. What is the probability they will meet for lunch that day?
(b) They both arrive independently with Beta(a = 2, b = 2) distributed times, where the r.v. represents the number of hours after noon. What is the probability they will meet for lunch that day? Hint: it may help to break the calculation into 3 disjoint events.
(c) In 3 or fewer sentences, compare your results for the 2 parts above. Why does this make sense? Hint: it may help to sketch the marginal PDFs.