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Pi: Less Random Than We Thought
Autoversicherung writes "Physicists including Purdue's Ephraim Fischbach have completed a study comparing the 'randomness' in pi to that produced by 30 software random-number generators and one chaos-generating physical machine. After conducting several tests, they have found that while sequences of digits from pi are indeed an acceptable source of randomness -- often an important factor in data encryption and in solving certain physics problems -- pi's digit string does not always produce randomness as effectively as manufactured generators do."
... but it seems a shitty research, based on the article:
> Pi never scored less than a B on the tests, and in one case outperformed all the RNGs, which in addition to mathematical algorithms included a device that uses turbulence in a fluid as its source of randomness. But in most cases, pi lost out to at least one RNG, and in several it finished decidedly in the middle of the pack.
Obviously. There is no reason that pi would beat every RNG out there on a sample of numbers. It should just be slightly ahead the pack (if some RNG are bad), or just in the middle (if all are good).
> "Our work showed no correlations or patterns in pi's number set - in short, pi is indeed a good source of randomness," Fischbach said. "However, there were times when pi's performance was outdone by the RNGs."
Well, there is a reason why mathematicians consider that statistics are not a branch of mathematic. And such article are a proof of it.
pi output on the statistical tests were correct (if they werer not, then it would be an important news, as it would imply correlations). The fact that some other RNG generated "better" output for the (relatively) small sample they used is meaningless.
As far as I have read, this has yet to be proven.
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It's not that Pi is random or was ever though to be. But you can generate random (or not so random according to the article) numbers by picking out single digits from Pi.
So I could take, for example, every 14th digit in Pi and that would make a good random string of numbers between 0 and 9.
That is a true and fun little fact, but it is nothing special to pi. You can do that with any irrational number, i.e. sqrt(2). Anyway, this story is ridiculous, noone pay attention to it. They did (from the article) 2 or 3 tests, the most significant appearing to be dividing 100 million digits into blocks of 10, plopping a decimal in the front. They then grabbed these blcoks in groups of 3 for x,y, z coordinates. They mapped these points in an imagnary cube and then graphed their distribution in the cube. From this they concluded that the other RNGs are more random. That is an extremely false conclusion. Arguing that one distribution is more random simply because it covers more of the cube or it's distrbution is more of a bell curve is just plain stupid so I really hope I missed some important fact when I read the article. Random is random and there is no rule saying that randomness is only random if it is distributed evenly or forms a bell curve (any such constraint would go against the nature of being random). Most RNGs try to distribute digits in a even manner because for cryptography purposes it is important, but is pointless when trying to deal with true sources of randomness. The fact that there is any such predefined distribution obviously shows that it isn't random (thus they are called pseudo-random), but arguing one algorithm generates a bell curve and another doesn't so the first one is better is just a dumb argument when dealing with random numbers. I hope a few mathematicians chime in and either blow my argument out of the water or confirm what I said.
Regards,
Steve
I don't see why one should expect Pi to be the ultimate in mathematical random number generation. Its chaos comes from the fact that it is an iterative function; why should we assume that this particular iterative function generates more chaos than others? That would be too convenient.
Dear Lord: One of your creatures may be hurt tonight. Please let it be the other creature.
The real issue with statistics is that people who use them generally do not understand them. I get irritated with people all the time when people "prove" some statement. Statistics shows that a sample of the populace has some correlation within some bound that is likely to be true some percentage of the time. So, the real question is: what was the bound and what percentage of the time was the randomness within that bound. If PI's bound exists outside of the statistical error of the bounds of the other tests then one could say that PI is less random; however, it sounds like they indeed found a few tests where PI "beat" the other tests. In other words, the bound PI was within the statistical error of the other tests, but the computed mean was occasionally better. But, occasionally better is to be expected some percentage of the time. If it is with in that number of times, it is as you say, a meaningless conclusion. Statatics within the bounds of error are completely equal. Probability is math, but it is also just very probable that it is used wrong.
Bel, the mostly sane.. "Of course I can't see anything! I'm standing on the shoulders of idiots." -- Me
Pi is a transcendental number
Yes, that's right...
and therefore cannot be exactly determined
Er, that depends on what you mean by "exactly determined". Do we need to know the digits in decimal expansion (base 10) to "determine" pi? How about saying that pi is exactly "1.000" in "base pi"? IMHO, whether or not a number can be exactly determined is independent of whether its decimal expansion is known. By your logic, sqrt(2) cannot be exactly determined, as it is an irrational number and has infinitely many digits (and they aren't periodic, unlike 1/3=0.33333333333... which also has infinitely many digits). But I am not entirely comfortable with saying that sqrt(2) cannot be exactly determined. After all, we know exactly what it is -- the positive number whose square is two.
I expect e and the square root of 2 to be better choices
WTF? How is e a better choice? It is also a transcendental number, just like pi. And sqrt(2) isn't even transcendental!
This is untrue. The most common fallacy about random numbers is that they need to "appear" random.
Of the list of numbers,
734901253789
666666666666
123456789012
Which is random? One answer is that all of them may be random. There is no reason why 1234 is any less random than 7305. A truly random number with infinite digits will absolutely repeat any sequence of numbers you can think of of any length whatsoever.
Think of it this way: If you have a true random number generator, spitting out a digit every second, and you see it spit out:
1...2...3...4...
then can you predict what the next digit will be? If it is truely a random number generator, the answer is no, you can not. However, the next digit has a 1 in 10 chance (0..9) of being a 5, so it is possible. If you reject 1...2...3...4...5 as possible sequence, then you have instituted a rule restricting the possible outcomes of the random number generator--and have therefore reduced it's effective randomness. Rules defeat randomness, so 12345 is as valid a random number as any other sequence of five digits.
Jim