Here's a simple 'derivation' of the Brownian motion law that after N steps of unit distance 1, the total distance from the origin will be sqrt(N) on average. It's certainly not rigorous, but I'm wondering if people think it's reasonable, or possibly even a commonly known.
An object takes one step from the origin, so is at a distance 1: d = 1 for N=1.
On average, the next step will be neither exactly toward or exactly away from the origin, so you compromise and say it steps along a direction that's perpendicular to the vector connecting the origin to its present location - sort of half way between walking backwards and walking forwards. By Pythagoras's theorem, the average distance will then be d = sqrt(1^2+1^2) = sqrt(2) for N=2.
Likewise for N=3, stepping in a normal direction gives d = sqrt( sqrt(2)^2 + 1^2 ) = sqrt(3), so in general d = sqrt(N).
This seems to work in dimensions 2 or higher.