From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Why Gauss-Distribution? How Do We Know This?
Date: 9 Jun 1999 07:46:44 -0500
Newsgroups: sci.math
Keywords: How do normal (or Poisson) distributions arise in practice?

In article <7jkh71$68t$1@nnrp1.deja.com>,  <t_n_southton@hotmail.com> wrote:
>Gauss's Curve (for lack of a better word for it) is described by this:

>Y(X) = (2*Pi)^-0.5 * exp (-0.5*X^2)  .

>The area from -infinity to +infinity is 1 (even I can prove that!).
>However, how do we know that certain statistics are distributed in this
>fashion?  Certain stats which are Gaussian in nature are grade score,
>and height.  But how come they are not distributed as:

>Y(X) = (2/Pi) / (X^2 + 1)?  The area from -infinity to +infinity is 1
>in this curve as well, and heck, it even looks like a Gaussian curve.

It does not look TOO much like it.  But your question is good, and
the answer is that probably nothing in nature has the normal distribution,
which should not even be attributed to Gauss.

I believe it was Poincare who stated that "everyone" was assuming
normality; the theorists because the empiricists had found it to
be true, and the empiricists because the theorists had demonstrated
that it must be the case.

The distribution seems to have been discovered by de Moivre, in
approximating the binomial distribution using Stirling's formula.
This is a special case of the Central Limit Theorem, which states
that, under certain conditions, the sum of a large number of random
variables is APPROXIMATELY normal.

It was used as a distribution of errors by many, because it 
gives simple fits by maximum likelihood.  The reason is that the
logarithm of the density is a quadratic.  It is often the case
(the Gauss-Markov Theorem) that procedures based on it are good
even if it is not correct.

>What assumptions did Gauss make in fitting this curve to random
>numbers, and when are Gaussian distributions valid?

Gauss came up with many characterizations of normality, and did
use it to improve fits to orbits of objects within the solar system.  

 For example, if I
>did a frequency plot of the digits in Pi expanded out to a million, it
>probably won't be Gaussian, but a line with slope zero.

>Finally, what is a Poisson distribution?  From what I understand,
>here's an examlpe of a P.D:  say that you have an auditoreum filled
>with 1000 people.  All of a sudden, you throw in 1000 dollars in notes
>from the very top of the auditoreum and allow the people to collect as
>many dollars as they can.  The *AVERAGE* number of dollars per person
>is $1.  However, most people will have anywhere between 0-n number of
>dollar bills. This is Poisson Distribution.  What's the proof here, and
>what's the equation of a Poisson Curve?

A Poisson distribution is a discrete distribution; the formula is

	P(X = n | \mu) = exp(-\mu)*(\mu)^n/n|.

It is the limiting distribution of the number of occurrences of rare
events.

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558