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Statistics, more variance

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.