The problem framed this time is at least partially a classic problem in geometry. You can find a lot in the literature about where this general problem arises in numerous fields of application. Some previous exposure to spherical geometry would probably be helpful, particularly for part (e). Here is the problem; try to do as many parts as you can.

(a) Suppose $P, Q$ are two points on the $n$-dimensional unit sphere $\mathcal{S}^{n-1} = \{x \in \mathcal{R}^n: ||x|| = 1\}.$
Find the density of the Euclidean distance $PQ$ between $P,Q$. Draw a nice picture of simulations of the $PQ$ distance in dimensions $n$ = 2, 3, 5, 10, 25.

(b) Find the expectation of the Euclidean distance $PQ$ between $P,Q$.

(c) Is there something very interesting about the fourth moment of the distance $PQ$ when the dimension $n$ is even? What is it?

(d) Derive an asymptotic expansion for the expected distance $PQ$ of part (b). Explain intuitively why the leading term in the asymptotic expansion is what it is.

(e) Suppose $n$ = 3, and that you have to travel the shortest path from $P$ to $Q$ along the surface of $\mathcal{S}^{n-1} = \mathcal{S}^2$. What would be the expected distance travelled? Is this answer larger than the answer to part (b)?

### Solution to Puzzle 27

Denote $X_{(n)}, X_{(1)}$ by $Y, X$ respectively. By using the tailsum formula, one has $E(Y) = \theta + N – c(n,N)$, where
$c(n,N) = \frac{N-n}{n+1}$.
Thus, by symmetry, we get a linear unbiased estimate $aW+b$, where $a = a(n) > 1, b = -1$. This is indeed the UMVUE of $N$ by the Lehmann-Scheff\’e theorem, provided the parameters $\theta , N$ are arbitrary, i.e., $(\theta +1, N) \in \mathcal{N}\,\times \mathcal{N}$. The joint p.m.f. of $(X,Y)$ can be written too, from which the second moment of $W$ follows. Thanks to Andrej Srakar, PhD student in Mathematical Statistics at the University of Ljubljana, Slovenia [pictured left], for sending in a solution.