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MathGroup Archive 2004

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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.


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