## December 28, 2013

### The Bus Stop Problems

Since we had so much fun with Bayes Theorem in a recent post, I can’t resist another.

Young Economics whippersnapper Evan Soltas posed two problems to do with Bayesian probability:

- You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop as a Poisson process, with an unknown (to you) rate, $\lambda$. You don’t know $\lambda$, but say you have a prior probability distribution for it, $p_0(\lambda)$.
- What’s your expected wait time, $\langle T\rangle$, for the next bus to arrive?
- Say you’ve been waiting for a time $t$. What’s your posterior probability distribution, $p(\lambda)$, and what’s your new expected wait time?

- Let’s add some more information. Say that riders arrive at the bus stop via an independent Poisson process with an (unknown to you) rate, $\mu$. Whenever a bus arrives, all those waiting at the stop get on it. Thus, the number of people waiting is the number who arrived
*since*the last bus. Say you arrive at the stop to find $n$ people already waiting. You wait for a time, $t$, at which point there are $N$ other people waiting at the stop (i.e., $N-n$ arrived while you were waiting).- Given this data, what’s your posterior probability distribution, $p(\lambda,\mu)$?
- What’s your new expected wait time, $\langle T\rangle$?

These questions illustrate one of my favourite points of view on Bayes Theorem, namely that it induces a flow on the (infinite-dimensional!) space of probability distributions. Understanding the nature of that flow is, I think, the key task of the subject.

Infinite dimensions are hard to get an intuition for, so one of the first tasks is to cut the problem down to a finite-dimensional one.