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 $\frac{\vec{x}}{\Vert\vec{x}\Vert_2}$ and $\frac{\vec{x}}{\Vert\vec{x}\Vert^2_2}$

  1. In general.

  2. If $\vec{x} \sim \mathcal N_p(\vec{\mu}, \Sigma).$

  • 1
    $\begingroup$ If $x$ is uniformly distributed, the normalized one is is uniformly distributed on the unit hyperball. $\endgroup$ Jul 25, 2014 at 7:01
  • $\begingroup$ (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). $\endgroup$
    – whuber
    Jul 25, 2014 at 13:45
  • $\begingroup$ @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. $\endgroup$ Jul 25, 2014 at 17:46
  • $\begingroup$ @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 $[0,1]^d$. $\endgroup$
    – Glen_b
    Jul 26, 2014 at 7:46
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    $\begingroup$ @trendymoniker See my answer here:stats.stackexchange.com/questions/263896/… $\endgroup$
    – Henry.L
    Mar 7, 2017 at 3:20


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