Alice and Bob plan to arrive independently at a certain restaurant between 12:00 and 1:00. Suppose that the arrival times for each of them are independent, and distributed uniformly across the interval.
(a) What is the probability that Alice arrives before Bob?
(b) What is the probability that Alice arrives 10 minutes or more before Bob?
(c) What is the probability that the first one to arrive has to wait 10 minutes or more before the second one does?
(d) What is the joint cumulative distribution function that describes their arrival times, letting the value 0 represent arrival at 12:00, and the value 1 represent arrival at 1:00?
(e) Given that Alice arrives before Bob, what is the probability density function for Alice's arrival time? What is the probability density function for Bob's?
(f) What is the probability density function for the arrival time of the first person to arrive?