Re: Factoring, SF, and transforms

From: W. Dale Hall (mailtodhall_at_farir.com)
Date: 04/19/05 

Date: Mon, 18 Apr 2005 23:41:35 GMT

jstevh@msn.com wrote:
> Matt Gutting wrote:
>
>>jstevh@msn.com wrote:
>>
>>>Mike wrote:
>>>
>>>
>>>>In article <1113781981.399751.145370
>>>>@z14g2000cwz.googlegroups.com>, jstevh@msn.com says...
>>>>
>>>>
>>>>>Mike wrote:
>>>>>
>>>>>
>>>>>>In article <1113662846.563112.321750
>>>>>>@g14g2000cwa.googlegroups.com>, jstevh@msn.com says...
>>>>>>
>>>>>>
>>>>>>>Some posters have argued rather obsessively that you tend to
>>>
>>>get
>>>
>>>
>>>>>>>trivial factors, but how?
>>>>>>>
>>>>>>>Given that there are an equal number of trivial and non-trivial
>>>>>>>factors--an infinite number of each--how could you logically
>>>
>>>tend
>>>
>>>
>>>>>to
>>>>>
>>>>>
>>>>>>>get only trivial factors?
>>>>>>>
>>>>>>
>>>>>>For the exact same reason that:
>>>>>>Given that there are an 'equal' number of integers divisible
>>>>>>by 2 as there are divisible by 1000003 -- an infinite number
>>>>>>of each -- I am logically much more likely to get a number
>>>>>
>>>>>>from the first set if I choose an integer at random.
>>>>>
>>>>>>Can you explain where I have gone wrong with my reasoning?
>>>>>
>>>>>Sure, you seem to think you can pick an integer at random, but you
>>>>>can't.
>>>>>
>>>>>How can you randomly pick one integer out of infinity?
>>>>>
>>>>>The reality is that when people talk about picking at random, they
>>>
>>>mean
>>>
>>>
>>>>>over a certain range, like, pick a random number from 1 to 10.
>>>>>
>>>>
>>>>This could rapidly go off topic, but of course I can pick an
>>>>integer at random from an infinite set -- as long as I don't
>>>>require a uniform distribution. For example -- repeatedly
>>>>toss a fair coin until heads occurs for the first time. My
>>>>random (positive) integer is n : the number of tosses
>>>>required. Now I believe I have just chosen an integer at
>>>>random from an infinite set.
>>>>
>>>>Can you explain where I have gone wrong with my reasoning?
>>>
>>>
>>>
>>>Yes. There's an upward bound to your set, which is the total
>
> number of
>
>>>flips.
>>>
>>>There is no upward bound to the set of integers.
>>>
>>>You may argue that theoretically there's a probability that you can
>>>flip an infinite number of times and never get heads.
>>>
>>>Do you think that's a rebuttal?
>>>
>>>
>>>James Harris
>>>
>>
>>"Theoretically"? I thought we were talking about theory here; why are
>>you denigrating a result because it is true in theory?
>>
>>For any positive natural number n, there is a finite probability
>>that the first head results on the nth flip; and there is also a
>
> finite
>
>>probability that no head has resulted by the nth flip. Since it's
>>possible for any n that the coin hasn't landed heads by the nth flip,
>>there's no upper bound to the number of flips which could be made.
>>
>>Matt
>
>
>
> Except that, either the coin *does* eventually land on heads or it
> never does, which is the only possibility for n to go out to infinity.
>

So? We're talking about the choice of a single number out of our
distribution, or did you forget that? *Of course* every selection
of a positive integer will yield a finite number!

        *EVERY* INTEGER IS FINITE!

Once you've chosen a number, that is a single sample from the
distribution. A priori, it has no bound. A posteriori, surely
it is bounded, since it's already given. However, that's not
the question. There is a nonzero probability for selecting
any positive integer; this is what you were disputing.

> That is, if the coin lands on heads, then you have a number.
>
> So that's one set of cases:
>
> ALL cases where the coin eventually lands on heads.
>

For which one obtains an integer:

        Any positive integer is possible.

> Ok, so that leaves another case:
>
> The coin NEVER lands on heads, is the remaining case.
>

Right, the case for which one does not obtain a number.

The probability of this event is zero.

If you're generating random numbers, and the procedure you're
using doesn't yield a number, then you try again.

> Understand?
>

Apparently, you don't.

You have been arguing that it is impossible to pick a number
at random from the set of positive integers, unless an upper
bound is determined from the start. That is untrue, UNLESS
one also requires the distribution to be uniform. There are
plenty of integer-valued random variables. The example of
tossing a fair coin until heads shows up is one possibility:
it assigns positive probability to every positive integer,
viz

        P(X = n) = 1/2^n.

Granted, there are rare events: picking x = Graham's number
is exceptionally rare. However, that choice still has positive
probability.

>
> James Harris
>

One would think a mathematician of your (self-perceived) stature
should have understood this.

Dale.