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Re: Normal distribtion

  • To: mathgroup at
  • Subject: [mg49326] Re: Normal distribtion
  • From: "Roger L. Bagula" <rlbtftn at>
  • Date: Wed, 14 Jul 2004 07:29:32 -0400 (EDT)
  • References: <> <> <ccdlms$sd5$>
  • Reply-to: tftn at
  • Sender: owner-wri-mathgroup at

Abrupt peak due to product of Gaussian distributions:
Probably doesn't have Kurtosis near zero either.
Roger L. Bagula wrote:
> I found a better faster way to get a Gaussian/ white noise:
> In Mathematica notebook style:
> x[a_]=(1+Sqrt[1-a^2))/a
> Noise=Table[Exp[-x[Sin[2*Pi*Random[]]]^2/2/Sqrt[2*Pi],{n,1,500}]
> ListPlot[noise,PlotRange--> All,PlotJoined->True]
> It is a projective line ( circle to line random taken as the basic for a 
> normal distribution's amplitude.) based algorithm.
> Since it doesn't use either the polar method with has a choice loop or 
> the older 12 random method, the random operation has only to be used 
> once for each noise value.
> It is faster by several factors and can be translated to almost any 
> language.
> George Marsaglia wrote:
>>"philou" <philou2000 at> wrote in message
>>news:7228735a.0407050100.4695fc68 at
>>>I heard that it was possible to get a realisation of a normal
>>>distribution from two realisations of an uniform distribution. Can
>>>someone explain me how to do that ? What transformations should I do ?
>>Your hazy reference may have been based on my polar method for
>>generating a pair of independent standard normal variates X and Y:
>>If U and V are independent uniform in (-1,1), conditioned by
>>               S = U^2+V^2 < 1
>>then S is uniform in (0,1) and independent of the point
>>(U/sqrt(S),V/sqrt(S)), which is uniform on the unit circumference.
>>Thus if R=sqrt(-2*ln(S)/S) then
>>      X=R*U
>>      Y=R*V
>>are a pair independent standard normal variates, obtained by
>>projecting that uniform point on the unit circumference
>>a random distance with a root-chi-square-2 distribution,
>>exploiting the uniformity of S and its independence of
>>the random point on the circumference.
>>Of course one has to discard uniform (-1,1) pairs U,V
>>for which S=U^2+V^2>1, so each normal variate is produced
>>at an average cost of 4/pi=1.27 uniform variates.
>>This method is sometimes improperly attributed to Box and Muller,
>>who pointed out that pairs of normal variates could be generated as
>>rho*cos(theta), rho*sin(theta) with rho root-chisquare-2 , sqrt(-2*ln(U)),
>> and theta uniform in (0,2pi), a result we owe to Laplace, who showed us
>>how to find  the infinite integral of exp(-x^2) by getting its square
>>as the integral of exp(-x^2-y^2), then transforming to polar coordinates.
>>For a method faster than my polar method, requiring about 1.01 uniform
>>variates per normal variate, try the ziggurat method of
>>Marsaglia and Tsang, in volume 5, Journal of Statistical Software:
>>George Marsaglia

Respectfully, Roger L. Bagula
tftn at, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :

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