Mathematical riddle

take a room
measure distances in the room, all of them (theoritically an infinity for argument sake)
from zero meters to say 15 meters (again for argument sake)
how do u calculate the average measurement knowing that there is an infinity of measurements

One intuitive way to do this is to pick 2 random points, find their distance, and integrate over all possible pairs of points (with a probability associated with each pair).

integrate[distance(x1,x2)P(x1,x2),x1,x2], where
integrate[P(x1,x2),x1,x2] = 1
Since we want a uniform distribution, P = 1/L^2n

In the case of measuring distances on a 2 dimensional floor of size L x L,

integrate[sqrt[(x1-x2)^2 + (y1-y2)^2] (1/L^4),x1,x2,y1,y2, each from 0->L] =
L integrate[sqrt[(x1-x2)^2 + (y1-y2)^2] ,x1,x2,y1,y2, each from 0->1] =
(3 hrs on mathematica)
L(2 + Sqrt[2] + 5 ArcSinh[1])/15
~ .521405L