expectation of the distance between two points in a square

**nounuo** · 06-07-2012 09:05 AM

Hi all,

how can I find the expectation of the distance between any two points uniformly distributed in a square of side L.
I need a mathematical prove.

Thank in advance.

With best regards.

**BGM** · 06-15-2012 05:17 PM

Let $X, Y, U, V$ be 4 independent random variables with the identical distribution $\text{Uniform}(0, L)$ such that the points $(X, Y)$ and $(U, V)$ represent the Cartesian coordinates of the two required points which uniformly distributed in a square of side $L$ , with the bottom left vertex is the origin.

The Euclidean distance between these two random points are
$\sqrt{(X - U)^2 + (Y - V)^2}$

and thus the expectation is given by

$\int_0^L\int_0^L\int_0^L\int_0^L \sqrt{(x - u)^2 + (y - v)^2} \frac {1} {L^4} dxdudydv$

This is the most obvious formulation only. Not sure this integral have an explicit solution or not.

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