Re: Probability of picking a positive rational number at random

On Mar 14, 1:30 am, Tim Little <t...@xxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
On 2008-03-14, Ross A. Finlayson <r...@xxxxxxxxxxxxxxx> wrote:

A uniform probability distribution over the naturals would have that
the sum of a constant infinitely many times would equal one. Now,
that kind of notion is well reflected in that of arguments about the
differential, that the sum of infinitely many (constant) infinitesimal
width differential areas is a finite sum and exactly one.

Standard mathematics does not define integrals as sums of infinitely
many infinitesimal areas. It simply defines them as limits of finite
sums of finite areas, in cases where those limits exists.

Arguments have been presented here as to why the existence of a
uniform distribution over R_[0,1] implies a uniform distribution
over N_[0, oo).

You mean your own previous illogical arguments on the subject, I
assume.

http://groups.google.com/group/sci.math/msg/2e346cb26bc92f7d

Assumption verified.

then selecting a rational, similarly uniformly, for example a
censored uniformly random real from [0,1]

Whatever that is supposed to mean.

- Tim

For no finite difference is it the infinitesimal differential, and no
sum of finitely many of those is non-zero (except for example as
infinite spikes, for example as in Dirac's delta, not a standard real
function). The differential area is infinitesimal, and only the sum
of infinitely many of them returns the "correct" value, i.e., the
evaluation of integration of smooth integrable functions perfectly
matching the measure of area under those functions. The limit is the
sum. That's basically to the effect that integration is not different
than the infinitesimal analysis with the infinitesimal differentials.

My arguments are logical, and maintained arguments also quite
logically sound. They're for those with their own thoughts on the
matter.

For any uniform distribution over the naturals, EF (the natural/unit
equivalency function) is its CDF (cumulative distribution function).
(That might be axiomatic, casually, where of course there is a theory
with no axioms.)

Probability distributions of the set of integers are functions from N
to [0,1] such that the sum over N of f(n) equals one. So, a
probability distribution of N is a totally miniscule proportion of the
functions from N to [0,1], because for most of those, the sum over N
of f(n) is not equal to one.

Anyways given a well-ordering of distributions of N, sample a random
real (uniformly by infinitely many fair coin tosses censoring those
ending in all 1's) and then take the ordinal from a well-ordering of
the reals that maps to that random real, and use that ordinal to
indicate the particular distribution d of N corresponding to that
ordinal, in a well-ordering of distributions of N. Then, among the
values of N with maximal values d(n), sample one of those at uniform
random. That's a random natural integer no different than uniformly
random from all the natural integers. It produces an integer, and
there is no way to say which it is, only that each is as likely as any
other, because as a well-ordering of the reals exists without any
ordinal enumeration of its values, the integer exists. (The
parameters of statistical models are also unknowns, here the
parameters of these distributions are arbitrarily complex, and there
are generally infinitely many of them, in terms of their definition as
transforms of well-known standard, if non-uniform, distributions over
the naturals.)

Basically that has that by flipping a fair coin infinitely many times,
that a random real is sampled, where asymptotically, the real is
unique or Dedekind/Cauchy/Eudoxus isn't sufficient to encode all real
numbers, and, each other real has the same probability of being
sampled, in the symmetries of the infinite balanced (symmetrical)
binary tree. Sampling a real number in that manner of infinitely many
Bernoulli trials is not representative of a standard statistical
algorithm. Yet, without it, the probability of there being an
irrational element of the unit interval is decreased, with regards to
standardized distributions of the unit interval (standardized in the
probabilistic in scaling to the unit, not standard in the foundational
in denying infinite infinities and working with only real functions.)

Here there is some consideration that with half of the positive
rationals being less than one, and half of the positive rationals
being greater than one, in a probabilistic sense, that the median
value of positive rationals is one. There is no standard median of
natural integers, it would be "half infinity". So it could be
axiomatic whether a nonstandard "median" of the positive rationals was
one or "half infinity", in nonstandard probability theories.

Half of the integers are even. That's so for _all_ of them.

Ross

--
Finlayson Consulting
.

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