Now, since the variance is just \( \mathrm{var}(X_i) = \mathrm{cov}(X_i, X_i) \), we get the variance of the linear combination \( U = \sum_i a_i X_i \):

$$ \begin{equation} \mathrm{var}(U) = \sum_{i,j}a_i a_j \mathrm{cov}(X_i, X_j) \tag{11} \end{equation} $$

And in the special case when the stochastic variables are uncorrelated, the off-diagonal elements of the covariance are as we know zero, resulting in:

$$ \mathrm{var}(U) = \sum_i a_i^2 \mathrm{cov}(X_i, X_i) = \sum_i a_i^2 \mathrm{var}(X_i) $$ $$ \mathrm{var}(\sum_i a_i X_i) = \sum_i a_i^2 \mathrm{var}(X_i) $$

which will become very useful in our study of the error in the mean value of a set of measurements.