Covariance of Sums of Random Variables

Theorem
Let $n$ be a strictly positive integer.

Let $\sequence {X_i}_{1 \le i \le n}$, $\sequence {Y_j}_{1 \le j \le n}$ be sequences of random variables.

Then:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}$