I take the bus to work every day. Every bus has a serial number, but unlike in [the German Tank Problem](http://en.wikipedia.org/wiki/German_tank_problem), I don't know if they are numbered uniformly $1...n$.


Suppose the first $k$ buses are all different, but on day $k+1$ I take one I've been on before. What is the best estimate for the total number of buses?



Estimate population size based on repeated observation
Maximum likelihood estimate is the smallest $n$ for which
$$\left( 1+\frac{1}{n} \right)^k \leq \frac{n}{n-k+1},$$
that gives a value of $n$ asymptotically equal to $\frac{k^2}{2}$, consistently with the Birthday Paradox. Not sure whether an unbiased estimate would be better for any practical purpose; maybe you do have an a priori distribution for which a Bayesian estimate makes sense?


