# What is the Varinace of the sample variance of iid normal distributed variables

• May 10th 2010, 08:02 AM
pimponi
What is the Varinace of the sample variance of iid normal distributed variables
Hi Guys
I have a sample of N iid normal distributed variables. And I need to calculate the Variance of the Maximumlikelihood estimated variance (sample variance) $\displaystyle {\sum (X_i-\bar X)^2\over N}$

Can somebody help me. I just can't come to an answer

Cheers
Todor
• May 10th 2010, 03:32 PM
Moo
Hello,

This rv follows a $\displaystyle \chi^2(N-1)$ distribution :D So if you look at a table you should get its variance :p
• May 12th 2010, 11:14 PM
matheagle
Nope, the sum over $\displaystyle \sigma^2$ has a chi-square distribution.

$\displaystyle {\sum_{i=1}^n(X_i-\bar X)^2\over n}= \left({\sigma^2\over n}\right)\left({\sum_{i=1}^n(X_i-\bar X)^2\over \sigma^2}\right)$

$\displaystyle \sim \left( {\sigma^2\over n}\right)\chi^2_{n-1}$

Hence $\displaystyle V\left( {\sum_{i=1}^n(X_i-\bar X)^2\over n}\right)= V\left( {\sigma^2\over n}\chi^2_{n-1}\right)$

$\displaystyle =\left( {\sigma^4\over n^2}\right)2(n-1)$
• May 13th 2010, 01:47 PM
Moo
Quote:

Originally Posted by matheagle
Nope, the sum over $\displaystyle \sigma^2$ has a chi-square distribution.

I wonder where I wrote that it doesn't (Wondering)