Estimating the average building size Suppose you want to estimate the average building size (defined as the number of people living in the building) in a small city. You randomly sample N people from this city, and each person tells you the number of people living in his building (i.e. the size of his building). Assuming you don't know how many buildings are there in this small city, what would be your best estimator for the mean building size?
Note that it is possible for multiple people in your sample to come from the same building. For example, if there is a building with size 1000, it's possible that 800 people from this building are all in your sample, so we need to careful not to overestimate the number of buildings with size 1000. In fact, because of this, our sample will likely be over-represented by people from large buildings, even though there might exist many small buildings in this town.
I am quite clueless on how to proceed. Any ideas?

Suppose, the number of people living in a random building of the city (chosen uniformly) is $X$ and has probability mass function $P(n)$. Now the number of people living in the building where a random person (chosen uniformly) lives is $Y$ and has probability mass function $Q(n) = \frac{n P(n)}{\sum_{k = 1}^\infty k P(k)}$. From that it follows, that $\frac{Q(n)}{n \sum_{k = 1}^\infty \frac{Q(k)}{k}} = \frac{\frac{n P(n)}{\sum_{j = 1}^\infty j P(j)}}{n \sum_{k = 1}^\infty \frac{\frac{k P(k)}{\sum_{j = 1}^\infty j P(j)}}{k}} = \frac{P(n)}{\sum_{k = 1}^\infty P(k)} = P(n)$ and therefore $E[X] = \frac{1}{\sum_{k = 1}^\infty \frac{Q(k)}{k}}$.
Now, suppose $Q_N(n)$ is the empirical probability mass function of $Y$ (the fraction of respondents that lives in the houses with $n$ people).
Then $\frac{1}{\sum_{k = 1}^\infty \frac{Q_N(k)}{k}}$ will be a consistent estimator of $E[X]$.
And it is exactly the harmonic mean of the responses.
