From: "Dr. Michael Albert" <albert@esther.rad.tju.edu>
Subject: Re: on Gaussian...
Date: Wed, 23 Aug 2000 19:23:53 -0400
Newsgroups: sci.image.processing,sci.math,sci.physics
Summary: [missing]

On Wed, 23 Aug 2000 uttiramerur@yahoo.com wrote:

> why and what are the advantages of assuming something as Gaussian. I am
> working on an estimation problem and I notice that some unknowns are
> assumed to follow a normal distribution (?).

1) Gaussians are mathematically tractable.

2) The central limit theorm states that if a random variable
is the sum of a large number of random independent effects, none of which
dominate (there are technical conditions to assure this), then
the random variable will generally have a normal distribution.
For many purposes, one can argue that the conditions of this
theorm are met.  For example, if the random variable is 
the "jiggle" in the position of a star due to atmospheric
turbulence, one could argue that the jiggle is the approximately
additive sum of the jiggles induced by many distinct regions
of the atmosphere acting approximately independently.  Versions
of the central limit theorm are proven in advanced texts on
probability (again, they can differ in technical hypotheses
cheifly related to guaranteeing that the no single effect 
dominates).  The text by Feller is of note because it has
a relatively elementary proof which does not involve 
the theory of Fourier transforms.

3) Experience has shown that the Gaussian distribution often
gives very reasonable results.  Indeed, often the results don't
depend upon the distribution being exactly Gaussian, so long
as the mean and variance are "correct" and the tails aren't
too big.

But be careful--remember "Everyone believes in the normal 
distribution--the physicists believe it
is a mathematical theorm, and the mathematicians
believe it is has been experimentally verified."

Best wishes,
 Mike