TITLE: Average Distance Between Random Points in a Rectangle
QUESTION [11 upvotes]: 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?

REPLY [17 votes]: 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).