I'm trying to understand the intuition of this question:
Fred then visits Blunderville, where the times between buses are also 10 minutes on average, and independent. Yet to his dismay, he finds that on average he has to wait more than 1 hour for the next bus when he arrives at the bus stop! How is it possible that the average Fred-to-bus time is greater than the average bus-to-bus time even though Fred arrives at some time between two bus arrivals? Explain this intuitively, and construct a specific discrete distribution for the times between buses showing that this is possible
Is the key takeaway that the mean wait time for Fred can be severely skewed due to outlier events? Given the right parameters, if most of the buses take between 40 and 60 minutes to arrive, but there are two consecutive events that happen within 15 seconds of each other, it can skew the average wait time to make it seem like the mean is 10 minutes?
Yes, the fact that Fred's arrival time is more likely to happen in a big gap between buses is what can make Fred's average wait time longer than the average time between the buses.
If there are six buses per hour, and they all arrive within seconds of one another, then the average bus-to-bus interval is 10 minutes, but Fred almost certainly arrives within that hour-long interval, giving an average wait time of just below 30 minutes. If all of Blunderville's 144 daily buses arrive basically at the same time, then Fred's average wait time, assuming he could arrive at any time of day with equal probability, will be just shy of 12 hours, which is well above the 1 hour wait the problem asks for.