My cousin is at elementary school and every week is given a book by his teacher. He then reads it and returns it in time to get another one the next week. After a while we started noticing that he was getting books he had read before and this became gradually more common over time. Naturally, I started to wonder how one could estimate the total number of books in their library.
Say the true number of books in the library is $N$ and the teacher picks one uniformly at random (with replacement) to give to you each week. If at week $t$ you have received a book you have read before $x$ times, is there an unbiased estimator for the total number of books in the library and what is the variance of this estimator? Is there another biased estimator with lower variance?
In my cousin's case, in the first $30$ weeks he received a book he had received before $3$ times.