I would like to know how one can derive the pdf of some geometric quantity. My problem goes as follows:
Consider the hypersphere $\mathbb{S}^{n-1}$ (hypersphere in $n$ dimensions) and randomly choose $n$ points on that sphere. These $n$ points all lie on some hyperplane, intersecting this hyperplane with $\mathbb{S}^{n-1}$ points with the hypersphere yields a cap. We may rotate this cap such that it points towards $e_1$. Then the angular radius $r$ of this cap has distribution \begin{alignat*}{1} C_{n(n-2)}\sin^{n(n-2)}(r), for \, 0<r<\pi/2 \end{alignat*} with \begin{alignat*}{1} C_k = \frac{2\Gamma((k+2)/2}{\Gamma(1/2)\Gamma((k+1)/2)} \end{alignat*} This I learnt from the article "Expected number of vertices of a random convex polytope" by D.G. Kelly and J.W. Tolle. They cited R. Miles for that result ("Random points, sets, and tessellations on the surface of a sphere, Sankhya, Series A, 33(1971), 145-174) for this result but I cannot find a free version of it.
My approach for deriving this would go as follows: We may always suppose that the cap is symmetric around $e_1$. The radius of the cap is $\sin(r)^{n-2}$. So for all (independent) points to land on this specific strip, we have probability ${(\sin(r)^{n-2})}^n$. The $C_{(n-2)n}$ only serves to renormalize. How can I formalize this?
Best