Error propagation

If is a set of random variables with the covariance matrix C_x, and if File:Hepa img275.gif , File:Hepa img276.gif is a set of transformed variables with transformation functions which are linear or well approximated by the linear terms of the Taylor series

in the neighbourhood of the mean E(X), then the covariance matrix C_y of File:Hepa img278.gif is

where T is the matrix of derivatives ( Jacobi Matrix)

If the X_i are independent, i.e. if C_x is diagonal, the variances of the Y_i are given by the so-called law of law of error propagation