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

From: Brett (cauchy_1_at_yahoo.com)
Date: 10/11/04 

Date: 11 Oct 2004 10:31:47 -0700

The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote in message news:<waderameyxiii-229165.20571110102004@news.supernews.com>...
> In article <931f9c2.0410101841.97b1f13@posting.google.com>,
> cauchy_1@yahoo.com (Brett) wrote:
>
> > > The OP wrote "randomly selected points on a sphere" so there's really no
> > > ambiguity.
> >
> > My question is: What does "randomly selected points on a sphere" mean?
>
> I already answered this.
>
> > Maybe it is conventional to assign a uniform distribution on the
> > sample space.
>
> That's what it means.

Does this "convention" extend to all sets? What is a uniform
distribution on the natural numbers? (other than assigning P(E)=0 for
each subset E) Or is this convention restricted to say (appropriately
scaled) n-dimensional Lebesgue measure on compact n-dimensional
manifolds. Do you have any references?

> > But, that "interpretation" may not be what is desired.
>
> If it is not desired, then do not say it. Say something else instead.
>
> > For Bertrand's paradox, where chords
> > are randomly selected on a circle, the uniform distribution of points
> > on the circle does not satisfy scale and translation invariance.
>
> I do not understand what you mean (please explain),

I included a reference that explains.

> but even if it's true,
> so what? I'm not saying "sample space = circle, uniform distribution" is
> the best version of Bertrands' problem. I am saying that whatever version
> you wish to state, do so clearly and precisely. There is no paradox here
> beyond vague language.

You must be aware that probability theory has some application to real
world phenomena. So, given the problem of determining the
"probability" that a chord chosen "at random", whatever that means, on
a circle of radius R, has length greater than R(sqrt3) --- which
probability space is the best model for the problem? Obviously, once a
space is fixed, an "answer" can be easily obtained. Is it your
contention that a chord cannot be chosen at random? That would seem to
be an unnecessary restriction on the practical applicability of
mathematics.

- Brett

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