Independence of Mean and Variance of a Sample

Date: 12/14/2008 at 10:01:58
From: Tess
Subject: Independence of x(bar) and s^2 of a sample

I came across a very interesting fact in statistical theory that I
can't understand.  If a random sample is taken from a normal
distribution, the mean and variance of that sample are independent! 

The other very interesting fact that is even more logic-defying to me
is that this independence is only true for normal distributions. 

How can they be independent when they are from the same sample values?
Is there an easy way to understand a method of showing these to be true?

Date: 12/16/2008 at 08:54:10
From: Doctor George
Subject: Re: Independence of x(bar) and s^2 of a sample

Hi Tess,

Thanks for writing to Doctor Math.

Proving the independence of xbar and s^2 for the normal distribution
is an advanced topic.  You would probably need to consult an advanced
undergraduate level text for a full proof.  But here is an intuitive
look at it.

What question does xbar answer?  It tells us something about where the
distribution is centered.  What question does s^2 answer?  It tells us
something about the spread of the distribution.  These are two
different concepts, and each is separately and completely connected to
one of the two parameters of the distribution.  So finding that the
estimators are independent should not be a huge surprise.

As for other distributions, their first two moments are not associated
with separate parameters in the way that mu and sigma are for the
normal distribution.  Since the moments are more complicated functions
of the parameters, the estimates of the moments end up having some
kind of dependency.

These comments fall well short of any kind of proof, but hopefully
they give you some insight.  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/