Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere.

Intuitively, I know that that's because the probability of $X/\sqrt{X_1^2+\cdots+X_n^2}$ belonging to any region with the same area on the surface should be the same. But how can I prove it mathematically?

share|improve this question
See the references in the article. –  user64494 Jul 16 '13 at 5:30
@user64494 i saw that page before but the references there only mentioned the method, not a detailed proof. –  Julie Jul 16 '13 at 12:35
Have you tried to use spherical coordinates? –  Davide Giraudo Jul 16 '13 at 17:20
The proof in less than 600 characters. –  cardinal Jul 16 '13 at 23:25
@cardinal Thanks a lot! –  Julie Jul 17 '13 at 3:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.