Average Distance Between Random Points in a Rectangle

Solution 1:

The answer, given in (Burgstaller and Pillichshammer 2009; Mathai et al. 1999), 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 $d=\sqrt{L_w^2+L_h^2}$.

REFERENCES:

• Burgstaller, B. and Pillichshammer, F., "The average distance between two points", Bulletin of the Australian Mathematical Society, 80(3), pp.353-359, 2009.
• Mathai, A.M., Moschopoulos, P., Pederzoli, G., "Random points associated with Rectangles", Rendiconti del Circolo Matematico di Palermo, II, XLVIII (1999).