# Student Probability Seminar

#### The Inspection Paradox

**Speaker:**
Terrence Alsup

**Location:**
Warren Weaver Hall 805

**Date:**
Thursday, April 19, 2018, 11 a.m.

**Synopsis:**

*Suppose buses arrive on average every 10 minutes according to a renewal process . If you show up to the bus stop at some time , 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 is a Poisson process. We will also study the limiting behavior when . Finally, we discuss several practical implications of this paradox.*