Sums of uniform randoms

Math and Physics Programming

Started by Kaptein November 02, 2012 03:04 AM

10 comments, last by Kaptein 11 years, 5 months ago

Kaptein

2,226

Author

November 02, 2012 03:04 AM

This is my last resort after 2 days of devouring information from everywhere

My math skills are relatively poor though, but I have a feeling what I'm trying to do simply can't be done.

I am summing random variables that have uniform distribution, once they are summed they are of normal distribution
With only n = 3, the distribution is already extremely poor.
How can I, and is it even possible to make the distribution uniform again?

This is the ultimate example:

[source lang="cpp"]
for (1 ... N) {
X += uniform();
}

X = uniformize(X, N);
[/source]

http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/
the density function for my final distribution should be a constant 1/N

Inferiarum

739

November 02, 2012 08:57 AM

why not just multiply the variable drawn from the uniform distribution with N

Kaptein

2,226

Author

November 02, 2012 11:12 AM

that's just scaling, i need octaves to build a frequency
a real life example would be uniformly distributed whitenoise

Inferiarum

739

November 02, 2012 01:15 PM

I am not sure what you want to accomplish.

If you want a random variable with uniform distribution from 0 to N, and you have a variable uniformly distributed from 0 to 1 well then you just have to scale it with N.
I do not see the point in creating a random variable with some distribution and then wanting to transform it back to uniform, if you actually have a uniform distribution to begin with.

Bacterius

13,181

November 02, 2012 03:16 PM

I have a feeling you might be confusing the concept of random variable and probability distribution. Could you tell us exactly in detail what you are trying to do?

“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”

clb

2,152

November 02, 2012 03:36 PM

If you only have an API that allows you to generate uniform bits (0 or 1), you can generate a random number in the range [0, N] in the following way:



int uniform(); // Returns 0 or 1 uniformly at random.



uint32_t RandomNumber(int N)

{

   uint32_t NPow2 = largest power-of-2-number smaller or equal to N.

   int i = 1;

   uint32_t result = 0;

   while(i <= NPow2)

   {

      if (uniform)

         result |= i;

      i <<= 1;

   }

   return (uint32_t)((uint64_t)result * (N+1) / (NPow2 << 1));

}

However, this is most likely not the route you want to go through, unless you somehow have a very special setup where your RNG can only supply individual random bits. That sounds rare, and instead most RNG sources are built-in to produce random integers in range [a,b] and random floats in the range [0,1] (which equals roughly 23 bits of randomness).

Bacterius

13,181

November 03, 2012 02:39 AM

However, this is most likely not the route you want to go through, unless you somehow have a very special setup where your RNG can only supply individual random bits. That sounds rare, and instead most RNG sources are built-in to produce random integers in range [a,b] and random floats in the range [0,1] (which equals roughly 23 bits of randomness).

These special setups are not rare at all, though I agree you are more likely to find them in cryptographic settings (entropy pools) than in conventional PRNG's. Also, a small terminology nitpick: a random bit generator is called an RBG (and its pseudorandom counterpart is a PRBG).

What I would find rare is PRNG's directly producing floating-point numbers. As far as I can tell this is usually provided by a helper function which performs the floating-point division along the lines of random(max) / max. I recall a few purely floating-point generators, but in my opinion it's not a very smart way to go about it, integer arithmetic is very well-defined unlike floating-point arithmetic.

“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”

Kaptein

2,226

Author

November 03, 2012 03:06 AM

I asked a statistician, and I got a final answer: that i couldn't do this due to central limit theorem
When i sum random samples, I am destined to get a skewed probability distribution
nevertheless, interesting conversations above

alvaro

21,604

November 03, 2012 04:16 PM

Well, not if your uniform random distributions are not independent. The central limit theorem does not apply, and you can build a uniform distribution U(0,7) as the sum of seven U(0,1) distributions that happen to be identical.

But I still don't quite understand what you were trying to do, so this conversation feels kind of irrelevant.

Kaptein

2,226

Author

November 04, 2012 05:40 AM

i was using the randoms to create octaved frequencies for biomes
i didnt want noise, since gradiented noise has extremely uneven distribution (think of the derivatives on the extremities of a sphere)
i basically wanted to make saw-like linear frequencies

but...
http://www.johndcook...-random-values/

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