Re: Randomness


Matthias Klaey wrote:
"Matt Zellman" <matt.zellman@xxxxxxxxx> wrote:


mensanator@xxxxxxxxxxx wrote:
Matt Zellman wrote:
are the following string of digits random?

Not likely.


I suppose that was kind of a stupid question. Are they random by some
definition of random? and if so, what definition?


0101000000000000000000000000000010000100100000000110001000000000

0000110110111110001010100010011100110011011011011001100110100110

1101110011101111010000001101011100000111101111111101000110100110

0000000000000000000000000100000000010000000001000000000000001000

1010110100111010011010100010011001100010011010010001100100100110

1010100111100001010110111101000100001000010100101001000001011100

here's a better question. Could the preceding six strings of digits
conceivably be randomly generated?

Sure.

This question somehow reminds me of a theorem that I heard of, but
never actually seen: "For any given set of data there is a test such
that the data will pass the test" (or converse, "such that the data
will fail the test/such that the null hypothesis is rejected")
I believe it is attributed to Kolmogorov.
Does anyone know something more precise about this?

And Matt, by this theorem the answer to your firts question is "yes".
For the second: I have no idea. The best reference I can give you is
D.E.Knuth, The Art of Computer Programming, Volume 2, Chapter 3, Third
Edition, Addison-Wesley 1998, ISBN 0-201-89684-2. There you may learn
more about random sequences than you ever wanted to know :)

Greetings, Matthias Kläy

Thanks. The way the strings were generated is as follows:

I started with the first however many digits of pi (a number whose
digits are provably normal), and applied a series of tests to it:

the first sequence gives a 1 for every digit that is a 0 or 1
the second sequence gives a 1 for every digit greater than or equal to
5
the third sequence gives a 1 for every digit that is odd
the fourth sequence gives a 1 for every digit that is the same as the
previous digit
the fifth sequence gives a 1 for every digit that is greater than the
previous digit
the sixth sequence gives a 1 for every digit that is a 3,4,5, or 6

The resulting sequences are not (necessarily) normal, but I think they
can still be described as "random" as long as there is some nonzero
chance that a digit could be either a zero or a one.

.