What is the expected distance between two random points inside a sphere?

Use spherical polar coordinates, and w.l.o.g. choose the polar axis
through one of the points. Now the distance between the two points is

     sqrt ( r1^2 + r2^2 - 2 r1 r2 cos(theta))

and cos(theta) is (conveniently) uniformly distributed between -1 and
+1, while r1 and r2 have densities 3 r1^2 d(r1) and 3 r2^2 d(r2). Split
the total integral into two (equal) parts with r1 < r2 and r1 > r2, and
it all comes down to integrating polynomials.

More generally, the expectation of the n'th power of the distance
between the two points is

         2^n . 72 / ((n+3)(n+4)(n+6))

So the various means are:

     the (arithmetic) mean distance is  36/35      = 1.028571...
     the root mean square distance is   sqrt(6/5)  = 1.095445...
     the geometric mean distance is     2exp(-3/4) = 0.944733...
     the harmonic mean distance is      5/6        = 0.833333...
     the inverse root mean inverse square distance is
                                        2/3        = 0.666666...