Average angle of vectors Given the following set ($n$ dimensional vectors with the length $1$ where each component is positive):
$
S=\{x\in\mathbb{R}_{\geq0}^n: \|x\|=1\}
$
What is the average / expected angle between two of these vectors?
For the $1$-dimensional case it is trivial, but for the $2$-dimensional case it already seems to be hard to get this value.
Maybe someone can help me?
Thank you very much.

In two dimensional case, the problem becomes computing the mean of $|x-y|$ where $x$ and $y$ are independently drawn from a uniform random distribution in $[0, \frac{\pi}{2}]$, which is $$\left(\frac{2}{\pi}\right)^2\int_0^{\pi/2}\int_0^{\pi/2}|x-y|\,dx\,dy = \frac{\pi}{6}$$.
