How to solve $E(X_i \bar{X_n})$?

Question

I need to show that $S_n = \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X_n})^2$ is an unbiased estimator of $\sigma^2$, given a population of $N$ elements and $X_1, \ldots,X_n$ the, independent, identical distributed random variables, sample size that model the samples.

By using the linearity of expectations and the expression of variances in expectations I can solve this problem for a big part, the only problem is that I can't solve $E(X_i \bar{X_n})$. I suspect that this can be done by using a conditional expectation, but how does that exactly work?

Thanks for your time,

K. Kamal

Answer 1

$$E(X_i\overline X_n) =E\left(\frac{X_i}n\sum_{j=1}^nX_j\right)=\frac1n \sum_{j=1}^n E(X_iX_j).$$ You need then to find $E(X_iX_j)$. The answers for $j=i$ and $j\ne i$ will differ.