Suppose we randomly place 2 points the 100-dimensional unit sphere. So we have
$$x_1, x_2\in \mathbb{R}^{100}\quad\text{ and }\quad|x_1|=|x_2|=1$$
What's the expected value of the euclidian distance between them?
$$E[|x_1 – x_2|]=\ ?$$
From just eyeballing some data, the answer looks like $\approx1.2$
And what about in general? So for 2 points on an n-dimensional sphere?
Best Answer
Eyeballing it, choose coordinates so that one point is at $\theta=0$. Then $\theta$ is the only relevant angular variable, and the length of the chord is $2\sin{(\theta/2)}$ so the integral should reduce to $$ \frac{\int_0^{\pi} 2\sin{(\theta/2)} \sin^{n-1}{\theta} \, d\theta }{ \int_0^{\pi} \sin^{n-1}{\theta} \, d\theta} = \frac{2\Gamma(n)\Gamma((n+1)/2)}{\Gamma(n/2)\Gamma(n+1/2)} = \frac{2^n [\Gamma((n+1)/2)]^2}{\sqrt{\pi}\Gamma(n+1/2)} $$ using various trigonometric and Gamma identities. For $n=100$, this gives a large fraction that is approximately $1.412$.