Re: probability question for math-heads

["geeman@best.com" <geeman@best.com>]

Timothy C. May wrote:
> 
> It obviously depends on your definitions of period, gap, clusters,
> patterns, etc. These would have to be pinned down before a precise answer
> could be given. For example, if the sequence is 30 digits long, or 5 digits
> long, etc.

yes ... the more one looks at the "random sequence" question, the more
questions that arise!  Again, referring to Knuth ... anything that takes
as many pages in one of his books as "Definition of Random Sequence" does
is certainly nontrivial.

A "gap" in my current inquiry is the number of bit positions between occurrances
of a specific bit-pattern.  My current investigation uses same-bit repetitions (00, 11),
but of course that generalizes to any specific n-bit pattern (e.g. we should get
the same results **in a perfect random sequence** if we count gaps between the
pattern "11" or the pattern "01".

> 
> But the general _framework_ for looking at this problem would probably be
> the Poisson distribution. That is, if the _expected_ ("average") number of
> gaps, runs, whatever, is "m", then the number "s" of actual gaps/runs that
> is 0, 1, 2, 3, 4, ...., n, is given by the Poisson distribution:
> 
> P [s; m] = (exp (-m))  (m ^ s)  / (s!)
> 
> (I read this like this: "The Poisson probability of seeing s occurrences
> given that m are expected is e to the minus m times m to the s divided by s
> factorial." I literally remember the formula from this verbal pattern,
> having used it much in my earlier career.)
> 

I'm going to have to chew on this .... but thanks for the Poisson hint.  Actually,
it looks like some graphs I've generated just might have some similarity to Poisson
distribution!!!  Bears more thought.

I certainly appreciate your time in replying in such depth.  It'll take a few days to
digest.

Thanks again,
Regards