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Suppose you have $n$ random normalized vectors in the plane (i.e. all lie on the circle of radius 1).

What is the distribution of the angles (depending on $n$) between neighboring vectors?

Can we at least compute the mean of this distribution? What about its variance?

gota
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    I guess the right direction would be: points on circle can be described by angle $\theta$. So you have sample ${ \theta_{i} }{i=1}^n$ from Uniform[$0, 2\pi$]. The question is to compute distribution of difference between 2 consequtive order statistics: $\theta{(i)} - \theta_{(i-1)}$. (Where $\theta_{(i)}$ - ith order statistics). Moreover, it's known, that $p_{X_{(k)}, X_{(j)}}(x, y) = \frac{n \cdot (n-1)!}{(k-1)!(n-k)!} \cdot p_{X_{(k)}}(x) \cdot p_{X_{(j)}}(y) \cdot F_{X_{(k)}}(x)^{k-1} \cdot [F_{X_{(j)}}(y) - F_{X_{(k)}}(x)]^{j-k-1} \cdot F_{X_{j}}(y)^{n-j}, \ 1 \le k < j \le n$ – Joitandr May 15 '20 at 06:18

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