Consider the stochastic variables \( X_i \) and \( X_j \), (\( i\neq j \)). We have $$ \begin{align*} Cov(X_i,\,X_j) &= \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle\\ &=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle\\ &=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j \rangle + \langle \langle x_i\rangle\langle x_j\rangle\rangle \\ &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle + \langle x_i\rangle\langle x_j\rangle \\ &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle \end{align*} $$ If \( X_i \) and \( X_j \) are independent (assuming \( i \neq j \)), we have that $$ \langle x_i x_j\rangle = \langle x_i\rangle\langle x_j\rangle, $$ leading to $$ Cov(X_i, X_j) = 0 \hspace{0.1cm} (i\neq j). $$