Proof of Normal Distribution

Date: 10/15/97 at 07:08:30
From: Kevin Sauter
Subject: Proof of Sample Mean and Sample Std. Dev. being normal if 
sample is from a Normally Distributed Population

Dr. Math,

Can you explain (and prove) to me why samples taken from a normally 
distributed population will also be normally distributed?  Everyone in 
class agrees that it makes sense, but how can we show it?

If we have a population X and mean (mu) with Standard deviation 
(sigma), how can it be shown that the mean of the samples (X bar) will 
have a sample mean of mu X bar that equals mu, and a standard 
deviation of sigma Xbar equal to sigma divided by the square root of n 
(the sample size)?

Thanks a lot.

K. Sauter

Date: 10/15/97 at 09:51:47
From: Doctor Statman
Subject: Re: Proof of Sample Mean and Sample Std. Dev. being normal if 
sample is from a Normally Distributed Population

Dear Kevin,

There are three aspects to your question.

Consider the mean of a random sample of observations taken from a 
Normal population.

1. The sample mean, xbar, follows a Normal distribution.

   I would use moment generating functions to show this. I'm not sure
   what grade level you teach, but I would cover this in a 
   mathematical statistics course for junior or senior college math 
   majors. The bad news is that it is advanced, but the good news is 
   that if you do it this way, you will see that the mean is mu and 
   the standard deviation is sigma/root(n).

2. The mean of the distribution of xbar is mu.

3. The standard deviation of xbar is sigma/root(n)

   These last two can be built up using the following:

   Show E[aX+b] = aE[X]+b

   Show Var[aX+b] = a^2 Var[X]

   Show E[X+Y] = E[X]+E[Y]

   Show Var[X+Y] = Var[X]+Var[Y]+2 Cov(X,Y)

   Show Var[X+Y] = Var[X]+Var[Y] if X and Y are independent.

   Now you are ready to go for the main results:

   E[xbar] = E[(1/n) sumof xi] = (1/n) sumof E[xi] = (1/n) n mu = mu !  

   Cool!

   Var[xbar] = Var[(1/n) sumof xi] = (1/n)^2 sumof Var[xi] <- using  
     indpt.
             = sigma^2 /root(n) with a little algebra.

Hope this helps!

Sincerely,

-Doctor Statman,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/15/97 at 15:42:29
From: Kevin Sauter
Subject: Re: Proof of Sample Mean and Sample Std. Dev. being normal if 
sample is from a Normally Distributed Population

Danke schon, Herr Doctor!  Ich verstehe und ich mochte daruber weiter
sprechen mit mein schuleren.  

Auf weidersehen,

Kevin Sauter