Average distance between two points in a circular disk

See the answer to this question. The expected distance is $$ d= {128 r\over 45\pi}. $$

Here is another demonstration of this result.

With the probability density you can find the average of the distance, or the distance squared, or the variance of the distance, or whatever you want. The probability density for the distance $l$ between points in a circle of radius $r$ is given by Ricardo García-Pelayo 2005 J. Phys. A: Math. Gen. 38 3475 as $$p(l)=\frac{4l}{\pi r^2}\arccos \frac{l}{2r}-\frac{2l^2}{\pi r^4}\sqrt{r^2-\frac{l^2}{4}}$$

Then the average distance is given by $$\int_0^{2r}lp(l)dl=\frac{128r}{45\pi}.$$ The average of the distance squared is given by $\int_0^{2r}l^2p(l)dl=r^2.$ Thus the standard deviation is given by $$\sqrt{\int_0^{2r}(l-\frac{128r}{45\pi})^2p(l)dl}=\sqrt{r^2-\left( \frac{128r}{45\pi} \right)^2}=\frac{r\sqrt{2025\pi^2-2^{14}}}{45\pi}$$

Following Grimmett & Stirzaker 's "One thousand exercises in probability", we can approach this way:

Let $ f(r) := E \rho $, where $\rho $ is the distance between points P1 and P2 which are independently and uniformly distributed on the disk $D$ of a radius $r$.

Let's consider the problem for the disk of a radius $r + h$ for little $h$. If points distributed over a disk $D_h$ of a radius $r + h$: $$ P\{P1 \in D, P2 \in D \} = (\frac{\pi r^2}{\pi (r+h)^2})^2 = 1 - \frac{4 h}{r} + o(h) $$ $$ P\{P1 \in D, P2 \in D_h \backslash D \} = (\frac{\pi r^2}{\pi (r+h)^2})(1 - \frac{\pi r^2}{\pi (r+h)^2}) = \frac{2 h}{r} + o(h) $$ And finally: $$ P\{P1 \in D_h \backslash D, P2 \in D_h \backslash D \} = o(h) $$

Now let's rewrite $ f(r+h) $ using conditional expectation: $$ f(r+h) = E \rho = E(\rho 1_{P1 \in D, P2 \in D}) + 2 E(\rho 1_{P1 \in D, P2 \in D_h \backslash D}) + o(h) $$

It's easy to see, that $$ \frac{E(\rho 1_{P1 \in D, P2 \in D})}{P\{P1 \in D, P2 \in D \}} = f(r) $$ $$ E(\rho 1_{P1 \in D, P2 \in D}) = f(r)(1 - \frac{4 h}{r} + o(h)) $$

And also we can rewrite $$ E(\rho 1_{P1 \in D, P2 \in D_h \backslash D}) = \frac{E(\rho 1_{P1 \in D, P2 \in D_h \backslash D})}{P\{P1 \in D, P2 \in D_h \backslash D \} } (\frac{2 h}{r} + o(h)) $$ In which $\frac{E(\rho 1_{P1 \in D, P2 \in D_h \backslash D})}{P\{P1 \in D, P2 \in D_h \backslash D \} } $ is exact (with ac. $o(h)$) average distance between points on the disk of radius $r$ and on the edge. Which is easy to calculate in polar coordinates with the center on the edge of the disk. $$ \frac{1}{\pi r^2} \int^{\pi/2}_{-\pi/2} \int^{2r cos(\phi)}_{0} s^2 ds d\phi = \frac{32 r}{9 \pi} + o(h) $$

So we have: $$ f(r + h) = f(r)(1 - \frac{4 h}{r} + o(h)) + 2 (\frac{32 r}{9 \pi} + o(h))(\frac{2 h}{r} + o(h)) $$ $$ f(r+h) - f(r) = - \frac{4 h}{r} f(r) + \frac{128 h}{9 \pi} + o(h)*... $$ $$ f'(r) = - \frac{4}{r} f(r) + \frac{128}{9 \pi} + o(1) $$ With $f(0) = 0$ we get: $$ E \rho = f(r) = \frac{128 r}{45 \pi} $$

Some notes about the integral: disk scheme in geogebra

point F -- is the point in disk,

point C -- is the point on the edge,

distance between them is $s$, the angle between Ox and CF is $\phi$.

And since we transformed coordinates to polar, we should multiply by Jacobian matrix determinant, which is $s$.

And finally $ \frac{1}{\pi r^2}$ is density of point in the disk.