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)*References*: <7228735a.0407050100.4695fc68@posting.google.com> <QaednZQbSYcwpnTdRVn-vA@comcast.com> <ccdlms$sd5$1@smc.vnet.net>*Reply-to*: tftn at earthlink.net*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 >>news:7228735a.0407050100.4695fc68 at posting.google.com... >> >> >>>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 tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/

**Follow-Ups**:**Normal Product Distribution***From:*DrBob <drbob@bigfoot.com>

**Re: Re: Normal distribtion***From:*DrBob <drbob@bigfoot.com>