Re: Distance Between 2 Randomly-chosen Points on a Sphere
From: Brett (cauchy_1_at_yahoo.com)
Date: 10/12/04
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Date: 12 Oct 2004 11:53:19 -0700
The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote in message news:<waderameyxiii-94D588.18055311102004@news.supernews.com>...
> 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?
That is my whole point. The original problem was imprecise. 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).
- Brett
- Next message: Kaimbridge M. GoldChild: "The Parageodesic--The Genuine Great-Ellipse Distance"
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