Re: Distance Between 2 Randomly-chosen Points on a Sphere
From: The World Wide Wade (waderameyxiii_at_comcast.remove13.net)
Date: 10/12/04
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Date: Mon, 11 Oct 2004 18:05:53 -0700
In article <931f9c2.0410110931.5a4a838f@posting.google.com>,
cauchy_1@yahoo.com (Brett) wrote:
> 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?
No. It wouldn't make sense for every set.
> What is a uniform
> distribution on the natural numbers? (other than assigning P(E)=0 for
> each subset E)
There isn't one. That's why when someone comes on sci.math positing a
"random selection from the set of integers", he is given a warning, sent
out of the building, then allowed to come back in slowly.
> Or is this convention restricted to say (appropriately
> scaled) n-dimensional Lebesgue measure on compact n-dimensional
> manifolds. Do you have any references?
You have certain sets X that have well known and natural uniform
distributions on them. [0,1], S^n, O(n), any finite set, ... For such X the
language "randomly select" - without further qualification - almost always
refers to this distribution. That's what my mathematical experience tells
me, so that's my reference.
> > > 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?
This is a different question altogether. I do not address it, because I
have a finite time to live.
> Obviously, once a
> space is fixed, an "answer" can be easily obtained. Is it your
> contention that a chord cannot be chosen at random?
It is my contention that the phrase "choose a chord at random" - without
further specification - is seriously imprecise.
> That would seem to
> be an unnecessary restriction on the practical applicability of
> mathematics.
Just make it precise. If you can't do that, then what is this - literary
theory?
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