Derive mean and variance of x̄

Let $X_1, X_2, \dots, X_n$ be random variables, and define the sample mean as $\bar X = \frac{1}{n}\sum_{i=1}^n X_i$.

1. Assume the variables are i.i.d. with $\mathbb E[X_i] = \mu$ and $\mathrm{Var}(X_i) = \sigma^2$ . Derive $\mathbb E[\bar X]$ and $\mathrm{Var}(\bar X)$ .
2. Now remove the independence assumption. Derive $\mathrm{Var}(\bar X)$ in terms of the covariance matrix of $(X_1,\dots,X_n)$ .
3. As a special case, assume each variable has variance $\sigma^2$ and every pair $(X_i, X_j)$ for $i \ne j$ has the same correlation $\rho$ . Simplify the variance formula and explain how positive or negative correlation affects the precision of the sample mean.