1. ## Correlation of samples

Let $X_i, i= 1,..,n,$ be i.i.d. sampes from $N(\mu, \sigma^2).$ Let $\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i.$

Prove that $\bar{X}$ and $X_i - \bar{X}$ are uncorrelated for any $i.$

2. for simplicity let i=1....

Let's obtain the covariance between $\bar X$ and $X_1-\bar X$

$Cov(\bar X,X_1-\bar X) ={1\over n} Cov\left(X_1+\cdots +X_n,\left(1-{1\over n}\right)X_1-{1\over n}[X_2+\cdots +X_n]\right)$

Now find the covariance between each pair, taking one term from each set.

ALL we need is for the $Cov(X_i,X_j)=0$ for each $i\ne j$ , we don't need NORMALITY...

$\left({1\over n}\right)\left[\left(1-{1\over n}\right)\sigma_1^2-{1\over n}\left[\sigma_2^2+\cdots +\sigma_n^2\right]\right)$

Next use the fact that all the variances are equal this becomes zero.
Changing 1 to i is easy, just sum over all the terms that's not i in the second sum.