# Expectation of sample variance

Hi all, Just a quick question - I want to make sure I'm not missing anything obvious here! I'm trying to evaluate $E(S^2 \mid \bar{X} = \bar{x})$, where $X_1,\ldots,X_n$ are i.i.d. Normal($\mu, \sigma ^2$) random variables, $\bar{X}$ is the sample mean ($\frac{\sum{X_i}}{n})$ and $S^2$ is the sample variance ($\frac{1}{n-1}\sum{(X_i-\bar{X})^2}$).

As far as I can tell, this should just be $\sigma^2$, which is the expected value for sample variance (since its an unbiased estimator). Am I missing something though? Does the $\bar{X}=\bar{x}$ condition have some effect that I've not noticed?

This is not research level. Nevertheless, under your assumptions, the empirical mean and variance are independent so that $E(S^2|\bar{X}=\bar{x})=E(s^2)$. –  Jochen Wengenroth Feb 19 '13 at 7:14