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MathGroup Archive 2004

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Re: Normal distribtion

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49224] Re: Normal distribtion
  • From: "Roger L. Bagula" <rlbtftn at netscape.net>
  • Date: Thu, 8 Jul 2004 02:50:59 -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>
  • Sender: owner-wri-mathgroup at wolfram.com

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.

Ray Koopman wrote:
> "Roger L. Bagula" <rlbtftn at netscape.net> wrote in message news:<ccdlms$sd5$1 at smc.vnet.net>...
> 
>>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.
>>[...]
> 
> 
> (1+Sqrt[1-a^2])/a = Cot[ArcSin[a]/2], so
> y = x[Sin[2*Pi*Random[]]] = Cot[Pi*Random[]] has a Cauchy distribution.
> 
> Exp[-y^2/2]/Sqrt[2*Pi] is the standard normal density function, 
> but why do you use it here?
> 


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