Random numbers need not be uniform

http://en.wikipedia.org/wiki/Benford%27s_law

25 32 16 years ago

en.wikipedia.org

jgrahamc3• 16 years ago

The phrase "Random numbers need not be uniform" is a really bad title for this. Random numbers are by definition uniformly distributed.

What you are referring to here is Benford's Law. Why not simply say that in the title?

ajuc5• 16 years ago

This is really a bad title, but random numbers indeed need not be uniformly distributed - for example Gauss distributed numbers aren't distributed uniformly (from definition:).

jgrahamc3• 16 years ago

I don't understand in what sense numbers drawn from a Gaussian distribution would be considered random.

fhars2• 16 years ago

Of course not. There are whole books on the subject: http://cg.scs.carleton.ca/~luc/rnbookindex.html

fhars2• 16 years ago

For what it is worth: here is an example of a non-uniform random variate:

  a = [0,0,0,0,0,0,0,0,0,0]
  1000.times do
    a[5 *(rand + rand)] += 1
  end

results in (actual numbers may vary randomly):

  irb(main):020:0> a
  => [28, 60, 96, 151, 171, 168, 146, 96, 63, 21]

snippyhollow• 16 years ago

I am disappointed, I thought you were about to link something stating that "it is not correct to always consider unknown random variables to be uniformly distributed" (take 1/X for instance of why this is wrong). Instead, Benford's law...

jellicle• 16 years ago

People are going to be confused by this, because the definition of "random" used by mathematicians and that used by programmers is different.

"Random" to mathematicians means the outcome of a probability experiment. If you flip a two-headed coin or roll a one-sided die, and record the results, the results are "random" - random within the set of possible outcomes, according to the function that generates them. So you dutifully record "heads", "heads", "heads" and "1", "1", "1" for your results. It's random!

"Random" for mathematicians inherently means "predictable". There's a known probability distribution.

The word "random" for programmers means something very different. It inherently means "unpredictable". The probability distribution must be flat across the space of possible outcomes. If we are operating in a base-10 system, the chance of any of the next digits occurring must be exactly 1/10.

My prediction is most of the discussion here will be people talking past each other, using different definitions of the word "random".

ajuc5• 16 years ago

I'm programmer - for me random string is a string, that is shorter than any programm that generates it (random string has Kolmogorov complexity greater or equal its length).

I think this is the definition for mathematicians as well.

It's just that "random" is a common shortcut for - string comming out of probability experiment - it is very likely that it is random according to Kolmogorov complexity, because most strings are random, and space is very very big.

omaranto2• 16 years ago

While I think that while most mathematicians are aware of Kolmogorov/Chaitin complexity theory, the first thing that comes to mind when they hear "random" is the theory of probability. In probability a random variable is, roughly speaking, an experiment that has a set of possible outcomes each occuring with a certain probability. The probabilities are not necessarily the same.

I think most mathematicians would agree that Probability is a much larger and important branch of mathematics than Kolmogorov-Chaitin complexity, and usually use "random" in its probabilistic meaning.

scott_s2• 16 years ago

This phenomenon can be witnessed below, but, hey, people cleared up the confusion! Yay for civil discussions about technical topics.

Although I do disagree with the words you used to characterize the differences. While most people use "random" to mean "unpredictable," as you demonstrated, a uniform distribution is just as "predictable" as any other kind of distribution.

I think the real difference is that most people only associate "random" with uniformly random distributions. Since you likely need some formal probability education to be aware of the others, this isn't surprising.

sesqu2• 16 years ago

I don't agree with that. Yes, most people interpret the word "random" as "uniformly distributed" or "unpredictable", depending on context ("pick a number at random" vs. "some random glitch"), but statisticians and mathematicians interpret it as "nondeterministic".

That's not at all the same as "predictable", but the goal is usually to be able to characterize the phenomenon statistically such that certain properties, such as general location on the number line, can be identified. If such properties cannot be identified, that doesn't make the phenomenon any less random - just less understood and by extension less predictable.

ANH• 16 years ago

Speak for yourself. I am a programmer, and to me 'random' is a characterization of any variable drawn from a probability distribution, and it need not be uniform. In fact, most of the random time series I generate are produced by drawing from non-uniform (e.g. power law, gaussian) distributions. And I think calling them "predictable" is a bit of a stretch. They are predictable only insofar as I can say, "This variable will probably not go above or below X."

Perceval• 16 years ago

>and "1", "1", "1" for your results. It's random!

That's the problem with random, you can never be sure: http://www.dilbert.com/dyn/str_strip/000000000/00000000/0000...

p3ll0n• 16 years ago

Benford's Law is a special case of Zipf's Law (http://en.wikipedia.org/wiki/Zipfs_law) as they both originate from the same scale invariant functional relation from stat mech.

pohl2• 16 years ago

Sweet...a wikipedia page that reliably causes an "Aw, snap" in Chrome (5.0.375.70 on MacOS X). Is it just mine?

(In case anybody is wondering whether I've done the right thing and reported the problem: yes, I have.)

scott_s2• 16 years ago

Strange. Loads fine for me. Same version of Chrome on 10.5.8.

pohl2• 16 years ago

Interesting. I'm on 10.6.3. I'm also using the 64-bit kernel, which isn't the default. I wonder if one or both of these variables accounts for the difference.

Random numbers need not be uniform

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