I am wondering if the community could give me some pointers on how to solve the following problem. I feel as if I am missing a key link and so I am having some challenges making some headway. Any Assistance would be greatly appreciated. Here we go!
There are $N$ fishermen on a lake with a fixed, yet unknown, number of fish in the lake. Each fisherman fishes at the same time and for a length of time $T$. When a fisherman catches a fish, the fisherman annotates the time when the fish was caught. There is no catch and release.
At the end of the fishing period, a person collects all times, sorts them and computes the inter-catch (inter-arrival) times. He notes that they are power-law distributed according to
\begin{equation} \frac{\beta \alpha^\beta}{x^{1+\alpha}} I_{[\alpha,\infty)}(x) \end{equation}
where $I_{[\alpha,\infty)}(x)$ represents that indicator function, or domain, where the distribution is defined.
With this setup, is it possible to estimate the population of fish within the lake?