The Inspection Paradox (Bus Waiting Time)

Buses arrive at a stop according to a Poisson process with rate $\lambda$ (so inter-arrival times are iid $\text{Exp}(\lambda)$ with mean $1/\lambda$). You arrive at the bus stop at a uniformly random time, independent of the bus schedule. Let $L$ be the length of the inter-arrival interval that contains your arrival time — i.e., the time between the last bus before you arrived and the next bus after. (a) Find $E[L]$. Explain why it is not $1/\lambda$ despite inter-arrival times having mean $1/\lambda$. (b) Find the expected waiting time $E[W]$ until the next bus, where $W$ is the time from your arrival until the next bus. (c) A city official surveys bus riders and asks how long they waited. If the reported average is $1/\lambda$, should the transit authority be surprised? Explain using the inspection paradox.