Perhaps I should have used 'typical'? Your argument works in any dimension, but the % of small angles falls off pretty rapidly. Let me define 'equator' as those points on the hypersurface with angle pi/2. Consider the 'band' within +- 5 degrees of the equator. What we want is the ratio of the *hypersurface area* of this band to the hypersurface area of the entire hypersphere. BTW, does Archimedes' 'hat box' theorem work in dimensions other than 3 ? At 09:15 AM 10/27/2020, Tom Duff wrote:
Picking random vectors is problematic -- what's the distribution? If we settle on random unit vectors, wlog we can make one of them (0,0,0,1). Then the cosine of the angle between them is the w coordinate of the other. Positive and negative w's are equally likely, and (arccos(w)+arccos(-w))/2 = pi/2 for all -1<=w<=1, so the average is pi/2.
On Tue, Oct 27, 2020 at 12:02 PM Henry Baker <hbaker1@pipeline.com> wrote:
Pick 2 random vectors in 4D.
What is the angle between them, in maximum likelihood ?