pr.probability - Covariance of points distributed in a n-ball

The covariance is a multiple of the identity by simple symmetry considerations. For the constant, you just need, again by symmetry, and integration in spherical coordinates,

$$mathbb{E} X_1^2 = frac{1}{n} mathbb{E} |X|^2 = frac{c}{n} int_0^R r^{n-1} r^2 dr = frac{c}{n(n+2)}R^{n+2},$$
where $R$ is the radius of your ball and $c$ is a constant depending on $n$ and $R$. To identify $c$,
$$1 = c int_0^R r^{n-1} dr = frac{c}{n} R^n,$$
so $c = n/R^n$ and your covariance is $frac{1}{n+2} R^2$ times the identity matrix. Hopefully I've included enough detail that if I've made an algebra mistake it will be easy for someone else to correct it, but I think I recognize that as the right answer.

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