If \( X_i \) and \( X_j \) are independent, we get \( \langle x_i x_j\rangle =\langle x_i\rangle\langle x_j\rangle \), resulting in \( \mathrm{cov}(X_i, X_j) = 0\ \ (i\neq j) \).

Also useful for us is the covariance of linear combinations of stochastic variables. Let \( \{X_i\} \) and \( \{Y_i\} \) be two sets of stochastic variables. Let also \( \{a_i\} \) and \( \{b_i\} \) be two sets of scalars. Consider the linear combination:

$$ U = \sum_i a_i X_i \qquad V = \sum_j b_j Y_j $$

By the linearity of the expectation value

$$ \mathrm{cov}(U, V) = \sum_{i,j}a_i b_j \mathrm{cov}(X_i, Y_j) $$