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Topic: the probability theory has holes!
Replies: 31   Last Post: Dec 3, 2009 7:55 PM

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 ostap_bender_1900@hotmail.com Posts: 681 Registered: 2/1/08
Re: the probability theory has holes!
Posted: Dec 3, 2009 7:55 PM

On Dec 3, 8:53 am, James Burns <burns...@osu.edu> wrote:
> Ostap S. B. M. Bender Jr. wrote:
>

> > On Dec 1, 11:39 am, James Burns <burns...@osu.edu> wrote:
>
> >>One of the more pleasant surprises on the Internet
> >>is finding someone who will admit that they were wrong
> >>(when they finally understand that they were wrong).

>
> >>I wonder, are you going to surprise us?
>
> > I have a surprise for him: the uniform measure on Z+
> > (or any other infinite countable set) doesn't exist,
> > so there can be no valid probability arguments
> > of this sort.

>
> I suppose I understand what you're saying here:
> There is no function f: Z+ -> R such that both
>     f(m) = f(n) for all m,n in Z+
> and
>     Sum_{n in Z+} f(n) = 1
> That's easy enough to prove. Certainly, I am not
> disputing that.
>
> However, it looks to me as though a good argument
> very much like the quick-and-dirty arguments
> being used in this thread is possible with
> an expanded definition of a measure on Z+,
> a definition much like that for generalized
> functions, with the Dirac delta function as a
> particular example.
>
> As I understand it, the Dirac delta is described
> as a function that is 0 everywhere except one
> point, but, when it is integrated across that
> point, we get 1. This is obviously (and provably)
> impossible, as it is stated.
>

Well, I suppose you can define some relevant probability measures in
the realm of functional analysis, where the the Dirac deltas live.

>
> The work-around that
> I am familiar with uses a sequence of
> functions < d_n > such that
>     d_n(x) -> 0, as n -> infinity, for x <> 0,
> and
>     INT{-inf, +inf} d_n(x) dx  =  1, all n
> There are a lot of choices for such a sequence.
>
> A rigorous argument involving Dirac deltas
> would solve some problem using a convenient
> choice for the d_n, get an answer involving
> n, and then take the limit as n -> infinity
> as the "true" answer. The operations behind
> what we call the Dirac delta are completely
> unexceptional, but they have the same effect
> as the provably impossible function.
>
> My question is: why can't something similar
> be done to create a uniform measure on Z+,
> or, at least, something that would give us
> the answers a uniform measure would, if such
> a thing existed?
>
> For concreteness, suppose
>     f_n : { m in Z+ | m =< n } -> R
> where
>     f_n(m) = 1/n
>
> Suppose we ask what the probability that a
> "random number" is not divisible by 2, 3, or 5
> (for example), and get a constant plus some
> bounds that -> 0 as n -> infinity. Without
> doing very much in the way of calculation, I
> feel confident that the constant would be
> (1/2)(2/3)(4/5).
>
> What is wrong with this argument?
>

Probably not much.

Howevewr:

"the probability theory has holes!"

2. How is this going to help one with the twin primes conjecture?

Date Subject Author
12/1/09 eestath
12/1/09 Kook Spotter
12/1/09 Bart Goddard
12/1/09 eestath
12/1/09 Bart Goddard
12/1/09 eestath
12/1/09 Jim Burns
12/1/09 Bart Goddard
12/1/09 Pubkeybreaker
12/1/09 Bart Goddard
12/1/09 Gerry Myerson
12/1/09 Bart Goddard
12/1/09 eestath
12/2/09 Dik T. Winter
12/2/09 Bart Goddard
12/2/09 eestath
12/2/09 Bart Goddard
12/2/09 eestath
12/3/09 Bart Goddard
12/2/09 eestath
12/1/09 Nick
12/2/09 Dik T. Winter
12/3/09 ostap_bender_1900@hotmail.com
12/3/09 Jim Burns
12/3/09 FredJeffries@gmail.com
12/2/09 Dik T. Winter
12/1/09 Henry
12/1/09 eestath
12/1/09 eestath
12/2/09 Richard Tobin
12/1/09 eestath
12/3/09 ostap_bender_1900@hotmail.com