Random Unit Vector From a uniform Distribution

[emob2p]

Hi,

I have encountered the following problem in my research. As I do not have a strong background in probability theory, I was wondering if anyone here could help me through the following.

I would like to know how one makes rigorous the problem of randomly choosing a unit n-dimensional vector from a uniform distribution.

This is like choosing an point on the n-sphere in which the problem can be solved by switching to generalized spherical coordinates. However, I have read that one can also generate a uniform distribution from a normal distribution of the vector's coorindates, and then dividing by the norm. It is not clear to me why this method produces a uniform distribution.

Thanks Much,
Eric

 

[D H]

The covariance matrix of a multivariate unit normal expressed in cartesian coordinates is a constant times the identity matrix. In other words, the multivariate unit normal has a spherical probability distribution. The direction is uniformly distributed over the unit n-sphere.

 

[emob2p]

In other words, the multivariate unit normal has a spherical probability distribution.

Why does this follow? And if so, how does one use the covariance matrix to obtain a probability density that can be integrated to find expectation values?