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".
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."
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.
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.
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.
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.
Comments
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".
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."
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.
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.
I was always confused by lack of definition of randomness in statistic/probabilistic meaning, so I've somewhat mixed these 2 branches of math.
Thanks for clearing this up.
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.
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.
>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...