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.