Given I know the area of the space that points can be in, the total number of points, and that a Poisson process generated the points, how can I find the average distance between any given point and its nearest neighbor?
This is what I have tried so far:
Let $\rho$ be the the area of the space over the number of points in the space, that is: $\rho = \frac{A}{n}$
Let $\alpha = \pi r^{2}\rho = \pi r^{2} \frac{A}{n}$ where $r$ is the radius of a circle.
Now since I am trying to find the average distance between any given point and its nearest neighbor the expected number of points that are within distance $r$ from the given point should be $1$: $$ 1=\sum_{k=0}^{n}k\left( \frac{e^{-\alpha}\alpha ^{k}}{k!} \right) $$ $$ e^{\alpha} = \sum_{k=1}^{n}\left( \frac{\alpha^{k}}{(k-1)!} \right) $$ at which point I got stuck.