       Re: Normal distribtion

• To: mathgroup at smc.vnet.net
• Subject: [mg49326] Re: Normal distribtion
• From: "Roger L. Bagula" <rlbtftn at netscape.net>
• Date: Wed, 14 Jul 2004 07:29:32 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Abrupt peak due to product of Gaussian distributions:
http://mathworld.wolfram.com/NormalProductDistribution.html
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 msn.com> wrote in message
>>
>>
>>>Hi,
>>>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:
>>  http://www.jstatsoft.org/index.php?vol=5
>>
>>George Marsaglia
>>
>>
>
>

--
Respectfully, Roger L. Bagula