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...