Darren New wrote:
>>     If I have a sequence derived from a Gaussian distribution, then it is
>> truly random by the way you defined it. Given the whole sequence, up to
>> a point, it doesn't tell me anything about what the next element could
>> be (other than what is obvious - that it follows a Gaussian
>> distribution).
> 
> Correct.
> 
>> However, for this sequence "every block of a particular
>> length occurs with equal probability" does not hold.
> 
> I think there's confusion conflating "random" and "normal" in our
> conversation about distributions here. Not every random sequence is
> normal, and not all normal sequences are random. Sorry if I confused
> "the kind of random you assume idealized monkeys will generate on
> keyboards" with "any sort of random distribution".

	Not sure I'm getting it. Just to clarify, I'm using the word normal in
the way you originally meant it (normal numbers). I'm using normal
sequence in the same sense.

	I was merely pointing out that you originally said:

"Given that truly random sequences are normal, and in a normal sequence
every block of a particular length occurs with equal probability"

	I was giving you an example of a truly random sequence that was _not_
normal.

> To have a "normal number", you need linear distribution, basically.
> I.e., all substrings of a given length appear with equal asymptotic
> probability as the number of symbols you look at gets large. *Given*
> that, every sequence of any given length will appear if you let N be
> infinity.

	Agree - other than with the use of the word "will".<G> What I mean is
that what you say is how I understand what probability theory says.

> In other words, in order to avoid Shakespeare appearing, you would have
> to have a non-random distribution *because* the sequence is infinite and

	I think I get the general idea, but to nitpick, you can avoid it even
with a random distribution. If I have a distribution where the
probability of typing the letter 'e' is forbidden, it's still random
(not "truly" random).

	I suppose you may object to my referring to it as random, but it is
consistent with probability theory: A uniform distribution from 0 to 1
is a valid random distribution - even though you've excluded all numbers
greater than 1.


> not just unbounded. In exactly the same way that 0*X is zero no matter
> how big X gets, until X is actually infinite.
> 
> (lim(x->inf) 0*X) =/= (0*inf)

	That's really not helpful. I get quite fussy when people use infinity,
and insist on a lot of rigor (which I may not understand<G>) before I
accept anything said by it. The LHS is _always_ equal to 0 (assuming x
is a real number). It's not equal to the RHS because the RHS is
meaningless. It's undefined.

-- 
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"


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