# Average distance between random points in 2D

## Main Question or Discussion Point

I'm looking for an analytical solution to a very simple problem I've come across.

Start with a circle of radius a. Now place n points at random positions inside this circle. Can you calculate the expectation value for the mean distance between the points?

For the sake of argument can you solve this for say n=5?

This problem is very simple to solve computationally by just distributing points, measuring the distances and looping many times but I am looking for an analytical solution.

Hi.

Distance d between the two points $(r_1,\phi_1) and (r_2,\phi_2)$ is
$$d(r_1, r_2, \phi_1, \phi_2 )= \sqrt{{r_1}^2 + {r_2}^2 - 2r_1 r_2 cos(\phi_1- \phi_2) }$$

The average distance is expressed as
$$\frac{1}{(\pi a^2)^2} \int_0^a r_1 dr_1 \int_0^a r_2 dr_2 \int_0^{2\pi} d\phi_1 \int_0^{2\pi} d\phi_2 d(r_1, r_2, \phi_1, \phi_2 )$$.

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It might be simpler to consider cartesian coordinates for this case. The idea would be the same as above.

Basically, you are doing the same thing as you would with any average, but this time it's infinite so you use an integral. Just sum the distance between every possible pair, then divide by the "number of possibilities" (the area squared).