I would like to compute the expected radius of a hypersphere (dimension $N$) given these conditions:
- radius $R\in[R_{min}, R_{max}]$,
- radius is acquired by uniformly chosing point from $S_N(R_{max})\backslash{S_N(R_{min})}$, and taking its radius,
- $A+B$ number of points $x = (x_1,x_2,...,x_N)$ have been chosen uniformly from the same subspace $S_N(R_{max})\backslash{S_N(R_{min})}$. $A$ points are from the space $S_N(R)\backslash{S_N(R_{min}})$, and $B$ points are from $S_N(R_{max})\backslash{S_N(R)}$,
- $E(R)=f(R_{min},R_{max},A,B,N)$.
Done so far: I have writted distribution function of $R$ as follows: $f_R(r)=\binom{A+B}{B}\frac{(r^N-R_{min}^N)^A(R_{max}^N-r^N)^B}{(R_{max}^N-R_{min}^N)^{A+B}}\text{ .}$ I've got the function by comparing volumes between hyperspheres for given radius. I've had the intention of trying to compute the expected value from this distribution function, but it gets a bit messy if I try to crack it and integrate polynomials that I would get.
Questions:
- (of very very minor importance) is general approach alright. I suppose it is.
- (of major importance) is there a nice way to compute the $E(R)$?
Probable way: If I would write $p(r,N)=\frac{r^N-R_{min}^N}{R_{max}^N-R_{min}^N}\text{ ,}$ then the density function would be like this $f_R(r)=\binom{A+B}{B}p^A (1-p)^B\text{ .}$ It looks rather similar to binomial distribution to me...