Uniform distribution on the n-sphere.

[MathematicalPhysicist]

Hi, 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.

 

[mathman]

You need to convert to n dimensional spherical coordinates. The "area" distribution should be uniform.