       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.
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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