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Let $\{X_r:1\leq r\leq n\}$ be independent $N(0,1)$ variables. Let $\Psi\in[0,\pi]$ be the angle between the vector $(X_1,X_2,\dots,X_n)$ and some fixed vector $R^n$. Show that $\Psi$ has density $f(\Psi)=(\sin\Psi)^{n-2}/B(1/2,n/2-1/2)$, $0\leq\Psi<\pi$, where $B$ is the beta function.

My idea is to apply the theorem from the book: if $X_1,X_2,\dots$ are independent $N(\mu,\sigma^2)$ variables, then $\bar{X}$ and $S^2$ are independent. $\bar{X}$ is $N(\mu,\sigma^2/n)$ and $(n-1)S^2/\sigma^2$ is $\chi^2(n-1)$. I don't know how to apply it. Any help is appreciated.

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    You may assume that the fixed vector is $(1,0,\ldots,0)$ and compute the cosine of the angle through a dot product. Do you know the distribution of $$\frac{X_1}{\sqrt{X_1^2+\ldots+X_n^2}}$$? – 2017-02-13
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    @JackD'Aurizio I don't – 2017-02-13
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    In such a case, find it and it will lead to the answer. – 2017-02-13

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