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I have the next RV:

$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$ where $$X_i \tilde \ N(0,1)$$ It's a random vector, and I want to show that it has a uniform distribution on the n-sphere with radius $$\sqrt{n}$$.

I understand that it has this radius, just calculate it. But I don't understand from calculating the CDF how to I arrive at uniform distribution.

Thanks in advance, MP.

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2 Answers 2

up vote 4 down vote accepted

You can prove that the $n$-dimensional Gaussian is invariant under transformation by $T$ for any orthogonal matrix $T$. (This is well known)

For orthogonal $T$ we have $\|TX\|=\|X\|$ hence $T\underline W = \sqrt n\frac{TX}{\|TX\|}$. Therefore the distribution of $W$ is also invariant under transformation by $T$

So the distribution of $W$ is invariant under any isometry of the sphere and the uniform distribution is the only distribution on the sphere that satisfies this condition.

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Thanks alot.... –  MathematicalPhysicist May 20 '13 at 12:24
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Just a hazy thought: once the radius question is settled, you have to prove that the direction of the rv is uniformly distributed. What about fixing an arbitrary unit vector $\vec{n}$, and computing the probability that $\langle \vec{n}, X \rangle \geq 0$? If it is $\frac{1}{2}$ for any $\vec{n}$, doesn't it give you the result?

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I don't think that's enough. Any distribution that's invariant under $x\mapsto -x$ will satisfy your condition. –  Tim May 20 '13 at 11:49
    
Indeed. $t\mapsto\mathbb{P}\{ \langle \vec{n}, X \rangle \geq t\}$ should be sufficient to characterize it, though. –  Clement C. May 20 '13 at 12:11
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You would need $t\mapsto \mathbb P\{\left<\vec{n},W\right>\geq t\} \mathbb P\{\left<\vec{n},X\right>\geq t\|X\|\}$. Which would be sufficient but hard to calculate. –  Tim May 20 '13 at 12:17
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