Brian Beesley wrote:
> On Monday 02 October 2006 21:15, Mike McCarty wrote:
>
>>[EMAIL PROTECTED] wrote:
>>
>>>Hi,
>>>
>>>Robert Betts said: "I meant to add that all the prime numbers are known
>>>to be distributed randomly--I repeat--randomly along the real line."
>>>
>>>Why "randomly" ?
>>>If I remember well, a set of numbers is random if the shortest way to
>>>describe it is to provide the list of these numbers (there is no
>>>algorithm to compute them, and knowing the first N numbers does not help
>>>to predict number numbered N+1).
>>
>>[snip]
>>
>>This is not any of the definitions of randomness I learned when I
>>was a grad student in Mathematical Probability and Statistics.
>
>
> Isn't statistical testing about the probability that a number of samples are
> drawn from the same population?
Umm, you aren't using the correct terminology, but that is the
essence of it. What you want is "likelihood", not "probability".
They are very different things, but without going into a long
explanation, it is difficult to explain.
> If I remember my statistics correctly, statistics doesn't actually define
> randomness itself, it merely describes a method for testing whether a finite
> sample of observations might have been drawn from a truly random infinite
> population, to a given confidence level.
Basically, yes. Populations don't have to be infinite. The term "random"
by itself is not defined. "Randomness" by itself is not defined.
"Random variable" has a very precise definition, however. "Random
process" has a very precise definition. "Random walk" has a precise
definition. So, there is no generally applicable adjective "random",
many things which we consider to be random have precise definitions
as composite terms.
> Do the digits in the decimal expansion of pi (3.14159...) constitute a random
> sequence? All known statistical tests appear to be passed at a high
By definition, they do not. The closest thing to what you are
suggesting is the Stochastic Process. But the digits of PI form
a degenerate Stochastic Process (using digit number as the "time"
index).
When a mathematician says so-and-so is a degenerate whatever, he means
that while *technically* it falls within the possible satisfaction of
the definition, it is not *really* what we mean.
Like a die with all sides marked with a "1". It always rolls a "1".
This is a degenerate Stochastic Process. So do the digits of PI, and
for the same reason. They are completely predictable. They are not
random.
To put it another way, they are not random for exactly the same
reason that pseudo random sequences generated by modular congruences
are not true random sequences (Stochastic Processes). As a general
rule, anything which can be generated by an Algorithm, or one could
say by a Computer Program running on a Digital Computer cannot,
by definition, be a random sequence. If such an object passes various
tests for Stochasticity, then it is called a "pseudo random sequence"
(or more precisely "Process").
> confidence level, if a sufficiently large sample is taken - yet the sequence
> is not random, any more than the digits in the decimal expansion of unity
> (1.00000...) are random, and for exactly the same reason - both 1 and pi are
> explicit unique values on the real number line.
You have missed the boat here. One does not "sample" the digits
of PI. It is an impossibility. There is no sample space. There
is no population of digits living out there to sample.
One can indeed take the first N digits of PI, and create histograms
and distributions of the digits. And, presumably, there is a limit
distribution as N->oo (though this is not guaranteed). This limit
distribution may or may not be uniform discrete. But one does not
and cannot sample the digits of PI. The sixth "sample" of PI is always
9. There is no randomness.
Mike
--
p="p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
You have found the bank of Larn.
I can explain it for you, but I can't understand it for you.
I speak only for myself, and I am unanimous in that!
_______________________________________________
Prime mailing list
[email protected]
http://hogranch.com/mailman/listinfo/prime
