Re: Generating "independent" random numbers

[Zelphir Kaltstahl]

On 10/3/23 20:25, Maxime Devos wrote:
> Op 03-10-2023 om 17:08 schreef Zelphir Kaltstahl:
> > Some time ago, I wanted to generate uniformly distributed floats though. 
> > There seems to be no facility in Guile to do that. So I took a look at 
> > Wikipedia: 
> > https://en.wikipedia.org/wiki/Normal_distribution#Computational_methods:
> >
> >  > An easy-to-program approximate approach that relies on the central limit 
> > theorem is as follows: generate 12 uniform U(0,1) deviates, add them all up, 
> > and subtract 6 – the resulting random variable will have approximately 
> > standard normal distribution. In truth, the distribution will be Irwin–Hall, 
> > which is a 12-section eleventh-order polynomial approximation to the normal 
> > distribution. This random deviate will have a limited range of (−6, 6).[55] 
> > Note that in a 
> > true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
> >
> > OK, for most purposes this seems good enough?
> That's _normal_, not uniform, like tomas wrote. 

Aye, sorry for that typo. Yes, my goal is normal distributed floats (leaving 
aside the finite nature of the computer and floats).

> Though, if you really want to (inefficient), you could apply the cdf of the 
> normal distribution to the normal random variable to get a
>
> First, I'll say that there is no such thing as an uniformly distributed float 
> (*), because the real line has length infinity.
>
> As such, you need to pick a bounded range from which you want to sample 
> uniformly.  For example, let's say [0,1), which can be rescaled as desired to 
> any finite half-closed interval.
>
> The uniform distribution on [0,1) has infinite support which makes things 
> difficult for computers, but it can be approximated by a distribution with 
> finite support, let's say
>
>    mu_N = sum(i=0..(N-1)) dirac(i/N)/N
>
> where dirac(i/N) is a Dirac-pulse situated as i/N and N is large.
>
> Up to scaling, this is simply the uniform discrete measure on {0,1,..,N-1}!
>
> So, to generate an (approximately) uniform random number on [0,1), you can 
> simply do
>
> (define (random-real)
>   (exact->inexact (/ (random N) N)))
>
> for a suitably large choice of integer N>0.
>
> (*) Ignoring for a moment there are technically finitely many floats, but the 
> uniform distribution on the discrete set of floats is likely not what you are 
> interested in.
>
> (I probably didn't have to go in so much detail but whatever ...)

That's what I get for writing the wrong word :D

OK that's some math to unpack, thank you!

Best regards,
Zelphir

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