# Student Probability Seminar

Suppose buses arrive on average every 10 minutes according to a renewal process $N(t)$. If you show up to the bus stop at some time $T$, then what is the amount of time you should expect to wait until the next bus? You might expect to wait half the length of the average interval, so 5 minutes, but you will actually end up waiting longer on average. This counter-intuitve result is known as the inspection paradox. In short, you are more likely to show up during a longer interval, so these longer intervals are over-sampled. We will prove this for a general inter-arrival distribution and give an explicit calculation for the case where $N(t)$ is a Poisson process. We will also study the limiting behavior when $t\to \infty$. Finally, we discuss several practical implications of this paradox.