Now, since the variance is just var(Xi)=cov(Xi,Xi), we get the variance of the linear combination U=∑iaiXi:
var(U)=∑i,jaiajcov(Xi,Xj)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:
var(U)=∑ia2icov(Xi,Xi)=∑ia2ivar(Xi) var(∑iaiXi)=∑ia2ivar(Xi)which will become very useful in our study of the error in the mean value of a set of measurements.