Are $X_i$ and $\bar{X}_n$ independent?

Question

Suppose $X_1,X_2,...,X_n$ independent random variables with an identical normal distribution $N(\mu,\sigma^2)$, and $\bar{X}_n=\frac{\sum_{i=1}^{n}X_i}{n}$. I could show that $\bar{X}_n$ has a $N(\mu,\sigma^2/n)$ distribution, but why is $X_i$ and $\bar{X}_n$ independent?

Answer 1

They are not. For example take $\mu=0,\sigma =1,n=2$. If $X_1$ and $\frac {X_1+X_2} 2$ are independent then Their covarinace must be 0. But the covariance is $\frac 1 2$.