Re: Distance Between 2 Randomly-chosen Points on a Sphere

From: Dave Seaman (dseaman_at_no.such.host)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 19:39:19 +0000 (UTC)

On 12 Oct 2004 11:53:19 -0700, Brett wrote:

> That is my whole point. The original problem was imprecise.

The problem was imprecise as to the meaning of "most likely", but there
was no imprecision in the specification of "randomly selected points on a
sphere". It's understood that "random selection" means "uniformly
distributed".

> What do
> you mean --- precisely --- when you say the points are "uniformly
> distributed" over the sphere? My understanding is that "uniform
> distribution" is a term applied to a random variable, and that random
> variables are measurable functions from a probability space into the
> set of real numbers. Again, I ask, what does it mean for points to be
> uniformly distributed over the sphere --- that is, taking the surface
> as the outcome space, and the Borel sets as the sigma algebra, 1) what
> is the probability measure, and 2) what is the random variable that is
> supposed to have a uniform distribution? The answer to (1) will affect
> the answer to (2).

(1) The probability measure is essentially Lebesgue measure on the surface
    of the sphere, suitably normalized (i.e., Lebesgue measure divided by
    4*pi*r^2).

(2) The random variable is a point on the surface of the sphere. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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