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Let $x,y\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_d)$ be two random $d$-dimensional standard normal distributed vectors and let $\theta$ denote the angle between them. ($I_d$ denotes the $d$-dimensional identity matrix).

Do you know about a probability distribution that models either

  • the distribution of $\theta\in[0,\pi]$, or
  • the distribution of $\cos(\theta)\in[-1,1]$, or
  • the distribution of $\frac{\cos(\theta)+1}{2}\in[0,1]$?

A paper (on page 18, second paragraph) says that the third option is distributed according to $Beta(d/2,d/2)$. Can you derive that? Or can you point me to a resource proving it? Or might it be that they're just using an approximation?

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