Re: Random Rational?

From: Keith A. Lewis (lewis_at_SPYDER.MITRE.ORG)
Date: 12/13/04 

Date: Mon, 13 Dec 2004 16:24:29 +0000 (UTC)

Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk> writes in article <cpjska$ibb$1@south.jnrs.ja.net> dated Mon, 13 Dec 2004 10:58:18 +0000:
>Tim Little wrote:
>
>> "The probability that a random rational number has an even
>> denominator is 1/3 (Salamin and Gosper 1972)."
>>
>> Are such concepts justifiable? It seems to me that there cannot exist
>> a reasonable probability measure on either of these (countable) sets.
>
>How about this:
>for each positive integer N let
>P_N = N^{-2}|{(m,n): m,n integers, 1<= m,n <= N, m/n has even denominator}|
>and let P = lim_{N->infinity} P_N (if that limit exists).
>
>Is P a good interpretation of "the probability that a random rational
>number has an even denominator"?

No. Or more precisely, nobody has shown it.

The distribution implied by your definition is not a uniform distribution
because it is weighted heavily toward low denominators. For example, m/n=1
happens N times in your m*n space, m/n=1/2 happens floor(N/2) times,
m/n=1/10000 happens floor(N/10000) times.

We all know that "uniform distribution" over a countably infinite set is
nonsense. Therefore, for the phrase "random rational number" to mean
anything, some other distribution must be specified.

>Is P = 1/3?

For what it's worth, I think the answer is yes.

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.