Is there a closed form expression for the covariance of a uniform distribution in a n-ball? I would like to develop a test for vector sums of points sampled from a uniform distribution in a n-ball. I need the covariance of the distribution to relate this to the CLT. I've taken a stab at it but I am not sure if I am correct.

Any pointers to statistics of uniform distributions in a n-ball would be very useful. Thanks.

  • $\begingroup$ just to check I've understood the terminology: by an n-ball you mean all points in ${\mathbb R}^n$ whose distance from the origin is at most some fixed constant, right? $\endgroup$ – Yemon Choi Aug 11 '10 at 21:06
  • $\begingroup$ (Not that there is anything wrong with your terminology, but I was momentarily uncertain whether you meant a $n$-polydisc.) $\endgroup$ – Yemon Choi Aug 11 '10 at 21:07
  • $\begingroup$ I mean an n-dimensional sphere with points distributed uniformly within it. I don't mean a uniform distribution on the surface. $\endgroup$ – drfrank Aug 11 '10 at 21:47
  • $\begingroup$ And sorry I wasn't clear earlier. :) $\endgroup$ – drfrank Aug 11 '10 at 21:48

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.