My question is similar to this one but for rectangles instead of lines.

Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed random points inside the rectangle, and why?


The answer, given here and here, is

$$ \frac1{15} \left( \frac{L_w^3}{L_h^2}+\frac{L_h^3}{L_w^2}+d \left( 3-\frac{L_w^2}{L_h^2}-\frac{L_h^2}{L_w^2} \right) +\frac52 \left( \frac{L_h^2}{L_w}\log\frac{L_w+d}{L_h}+\frac{L_w^2}{L_h}\log\frac{L_h+d}{L_w} \right) \right)\;, $$

where $d=\sqrt{L_w^2+L_h^2}$.

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  • 1
    $\begingroup$ It seems to be a simple problem. But it's not. Good find. (+1) $\endgroup$ – Patrick Li Oct 7 '12 at 14:18
  • $\begingroup$ The famous book "Integral geometry and geometric probability" of Santalo is available on line at (projecteuclid.org/euclid.bams/1183539854). $\endgroup$ – Jean Marie Oct 27 '17 at 10:55

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