Expected Waiting Time in a Poisson Process

This question is set in a bus-arrival scenario modeled by a Poisson process, where arrivals are spread randomly over time with a fixed average rate. The candidate is asked to reason about what a "random arrival time" means when the process has been running for a long time, and to determine the expected time until the next event and since the last event. It contrasts everyday intuition about regular schedules with the less obvious behavior of truly random arrivals, framed through waiting times rather than counting events.

The problem leans heavily on the exponential interarrival distribution and, in particular, its memoryless property and residual-life behavior. To handle it cleanly, a candidate should translate the story into interarrival-time distributions, then condition on arriving at an arbitrary time point in the stationary regime. Interviewers watch for recognition that the forward and backward waiting times are random variables with specific expectations, and for comfort working with continuous distributions, conditioning, and equilibrium or Palm-type arguments rather than naive "half the interval" reasoning.

What it tests

In any process where events occur according to a Poisson process, the times between events are independent and exponentially distributed. The exponential distribution's defining feature is its memorylessness: the probability of waiting an additional time for the next event is independent of how much time has already passed. This means that, regardless of when you observe the process, the expected time until the next event is always the same as the mean interarrival time. The same logic applies to the time since the previous event, because your random arrival is more likely to fall into longer intervals, making the expected 'backward' and 'forward' waiting times equal. This symmetry arises because, for exponential distributions, the expected residual life from a random point is equal to the mean interval itself, not half the mean as with fixed intervals.