Can a random number generator picking from a finite set exist?
Can a random number generator picking from an infinite set exist?

What criteria must be met for randomness to be achieved?

define randomness.

Define generator as well. Anyway, random number generators picking from a finite set do exist; the simplest of them is coin flipping (picks from "head,tail" set.) Dice rolling are not far more complicated and offer larger sets.

I find computers incapable of true randomness.

Yes, they do exist already and are very common.



Yes*, easy example is picking a random number 1-103 and if 100, 101, 102 or 103 come up you add, subtract, multiply or divide. That of course makes any integer possible.

*if you only allow a predefined set and pick one of the indices to determine your value then no, as it would take forever to generate the set and another forever to select a value.




I do not believe randomness can be achieved. That takes us to the nature of the universe. Some physicists believe that everything is a purely deterministic process whereas others believe everything is stochastic. While not a theoretical physicist, I adhere to the deterministic point of view (although that is the minority view). If the universe is not deterministic then yes, randomness could be conceivably achieved by somehow measuring the state of some random process. (again, some argue that measuring the state introduces bias and could potentially make it quasi-random at best).

Just assume (like with most computers) that you have a pseudo-ramdom number generator on the interval(0,1)

Can a random number generator picking from a finite set exist?
The answer is Yes. Simple example you map all sequence from (0-0.5) to 0 and the remainder to 1. You are picking from a finite set.

Can a random number generator picking from an infinite set exist?
The answer in real life is yes (radioactive decay). On a computer the answer is no because computers can only deal with countable and finite (albeit very large) sets. You can only approximate this process.

For example you can only store so many decimals and/or so many integers/digits based on, among other things the memory capcity of your computer.

Weather true randomness can be achieved is a philosophical question. If you believe in Baysian statistics then the answer is yes, because probability measure changes depending on the eye of the beholder. Even though a coin flip is not a random process but a completely deterministic outcome based on the force I apply to the coin, the angle and speed at which it hits the table, and other physics, I the observer DO NOT have that information so from my standpoint it is random.

In the 19th century, it was believed that all physics was deterministic and we simply lacked the ability to measure some processes. Some quantum theory states that certain physical processes are truly random (like certain kinds of radioactive decay). However this is neither proven nor disproven. Einstein did say that "god does not play dice".

Regardless of philosophy, the point to statistics is to estimate things based on an incomplete information set. Weather or not the process is "truely random" isn't critical as long as it is "random enough" and/or enough information isn't known. When statistical theory was redone in the 20th century, the definitions of everything were made very mathematically precise as to what exactly we mean by "random variable" and "stochastic process". And built up all theory from there. Statistics is (mostly) mathematically solid and is robust enough (if used correctly) to handle many real world applications without them completely fulfilling what is assumed.

@Detmer oddly a lot of statistical problems ended up looking like previously known physics equations although the methodology was developed independently. The solutions came faster because the physicists had already solved the equations in many cases:P
Black-Scholes equation was actually something in physics (can't remember what) that was known at least 20 years earlier for a completely different problem.

Ironically random.org does their random number generation by variations in atmospheric pressure ;)



My diff eq TA in college told me that. Its interesting all the stuff that has been independently developed throughout history.

yep. Also a lot of statistics was redeveloped using nothing but set theory initially. It's almost impossible to follow some of it. Many proofs were redone once they started reading physics and engineering journals I think:P
And yeah when I took a course in SDEs the prof would always spend about 10-15 minutes each lecture telling us the history of the stuff and how the fields ended up feeding off of each other and ended up "greatly simplifying" things by which he meant from 20 pages of set theory to 2 pages of complex differential equations and measure theory:P

isnt the theory just take a variable which changes then make the sample period small enough

and take only the least significant digit

then you basically have randomness

since your polling period cant be gamed to generate a desired response


in other words you poll your computers clock every now and then to use that as a random number generator seed

if you had other things you could measure small enough changes in such as atmospheric pressure mentioned above then you could always use more than one source if you felt some could be gamed

Measure Theory FTW. I had to take that fluff for my bachelors degree. impossible to follow my ass, they are just elegant :-P

"God does not play dice" or "God does not play dice with the universe" -- Albert Einstein

http://en.wikiquote.org/wiki/Albert_Einstein

no and no

You should all be discussing .999... = 1

why? the money data type doesn't do infinity.

enshula: the way it works is you have an algorithm that generates a sequence of numbers which is considered to be "pseudo random". These numbers are on the interval [0,1]. Then you take something like a system clock or some other factor and apply a formula to that info to determine where on the list you start generating numbers.
Some of the pseudorandom number generators are good enough to fool even the most stringent statistical tests (ie the sequence is "random" enough to give an accurate representation of a random variable).

From the uniform random number we can then generate any other random variable of our choosing which will be as "random" and as "good" as the uniform random number.

Mersenne twister is one of the best ones.
:)
http://en.wikipedia.org/...dorandom_number_generator
explains it well.

for industrial applications when making a large/complex simulation it is often useful to always start from the same place in the sequence to you can examine the effect of model changes as opposed to just "model noise".
in many cases one only needs a certain number of "randomly" generated scenarios.