Estimation of the expected Euclidean distance between two random points on a unit $n$-hemisphere

Question

What is the best approach to estimate, with Wolfram Mathematica, the expected Euclidean distance (in a $(n+1)$-dimensional space) between two points selected uniformly at random on a unit $n$-hemisphere?

The approach I have in mind uses an expression whose length is proportional to $n$, while I would like a simpler and more elegant approach.

Answer 1

Use a normal distribution to generate $n$ values and Normalize to get a point on the sphere. Make sure that the last coordinate always has the same sign using Abs. Generate millions of these points and estimate the mean distance between pairs:

n = 3;
topt[p_] := MapAt[Abs, Normalize[p], -1]
points = topt /@ RandomVariate[NormalDistribution[0, 1], {1000000, n}];
distances = EuclideanDistance @@@ Partition[points, 2];
Histogram[distances]
Mean[distances]

(* 1.13137 *)

Answer 2

Another way to do the sampling (taking advantage of the built in Sphere function and RandomPoint functionality (modified from a similar question on sampling from the surface of the sphere

distanceDistributionOnHalfSphere[dimensionality_, nSamples_:10^5] :=
  With[{
   (* take a few extra samples account for loss *)
   randomPointsOnSurfaceOfNSphere = RandomPoint[Sphere[dimensionality], {4*nSamples, 2}], 
   
   (* define an operator that deletes points when either last coordinate is negative *)
   upperHemisphere = DeleteCases[{{___, x_}, {___, y_}} /; (Negative[x] || Negative[y])]
   },
  
  (* apply operator to the list and compute list of distances *)
  EuclideanDistance @@@ upperHemisphere @ randomPointsOnSurfaceOfNSphere
  ]

(* Evaluate mean of the sample *)
MeanAround /@ distanceDistributionOnHalfSphere /@ Range[10]

(The $N=3$ result is in agreement with @flinty's result*)