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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?

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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
2  
    
@cardinal Thanks a lot! – Julie Jul 17 '13 at 3:14

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