Re: Normal distribtion

*To*: mathgroup at smc.vnet.net*Subject*: [mg49245] Re: Normal distribtion*From*: koopman at sfu.ca (Ray Koopman)*Date*: Fri, 9 Jul 2004 02:26:15 -0400 (EDT)*References*: <7228735a.0407050100.4695fc68@posting.google.com> <QaednZQbSYcwpnTdRVn-vA@comcast.com> <ccdlms$sd5$1@smc.vnet.net> <ccg4n7$ot0$1@smc.vnet.net> <ccis0d$49v$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"Roger L. Bagula" <rlbtftn at netscape.net> wrote in message news:<ccis0d$49v$1 at smc.vnet.net>... > Dear Ray Koopman, > I'm familar with the Lorentzian distribution also called Cauchy also > called classically the Witch of Agnesi. > What's your point in your trignometric identity manipulation to get > another of the "peak" distributions? > I use the Normal function to give my amplitude at the address on the > real line x :{ x,-Infinity, Infinity}. > By my experiments this function of mine gives a much larger variability > than the Mathematica built in White noise function or what you get from > a Polar normal distribution, but the logic of the derivation is clear: > 1) a random number is found in [0,1] > 2) a point on a circle is found > 3) that point is projected to the real line at x > 4) that real line value gives an amplitude of a distribution that is a > normal distribution. > > The result is a Gaussian normal noise. > I really can't make it any simpler. > The idea was to develop a Gaussian noise generator whose derivation was > simple and obvious. I think my problem is with your terminology. I reserve "normal" and "Gaussian" for variables whose density is proportional to Exp[-((x-m)/s)^2/2] for some location constant m and scale constant s. Variables whose density is close to that I might call "near-normal" or "normal-like", but the Cauchy distribution is so strongly-peaked and long-tailed that I would say it is far from normal. Indeed, Cauchy variables are often used to demonstrate how procedures that work fine with normal and near-normal variables can give terrible results with variables whose distributions are long-tailed. However, even if we accept the Cauchy as near-normal, I guess I also don't understand what you mean by "noise". I would have thought it would be y = x[Sin[2*Pi*Random[]]] itself, not f[y], where f is some probability density function.