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Does anyone know anything about the distribution of a random vector divided by it's norm? More specifically, if we know the distribution of a vector valued random variable $x$, what can we say about the distributions of $\frac{\vec{x}}{||\vec{x}||_2}$ and $\frac{\vec{x}}{||\vec{x}||^2_2}$

1) In general

2) If $\vec{x} \sim Normal_p(\vec{\mu}, \Sigma)$

trendymoniker
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    If $x$ is uniformly distributed, the normalized one is is uniformly distributed on the unit hyperball. – Marc Claesen Jul 25 '14 at 07:01
  • (1) is too general to obtain much useful information. (2) has been asked here, but unfortunately it would be difficult to search for that thread. I recall that an answer has been obtained but it is not a straightforward one except when $\Sigma$ is orthogonal and $\mu=0$ (which makes the normalized distribution uniform on the unit sphere). – whuber Jul 25 '14 at 13:45
  • @whuber: Thanks for the information. Any vague recollections on how to go about searching for the old threads? I'd be keen to learn more about this. – trendymoniker Jul 25 '14 at 17:46
  • @MarcClaesen "uniformly distributed" won't be sufficient on its own for that to hold. If it's uniformly distributed **in the unit ball**, then the normalized one would be uniform on the surface, but it wouldn't be true of $x$ uniform on $[0,1]^d$. – Glen_b Jul 26 '14 at 07:46
  • @trendymoniker See my answer here:http://stats.stackexchange.com/questions/263896/moment-mgf-of-cosine-of-two-random-vectors – Henry.L Mar 07 '17 at 03:20

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