Propagation of uncertainties for a function of observables
This is a personal derivation of a familiar result! Alex Read 14.05.2011
We want to estimate the uncertainty of a functionof a set of uncorrelated observables with associated uncertainties . We assume that the observables are sampled from normal distributions which are narrow relative to the form of the function and thus approximate the function with its Taylor-series expansion, keeping only the lowest-order derivatives , where represents the Gaussian-distributed deviation from the true mean (which is unknown, but estimated by the observed value) . In other words and .
The uncertainty inis estimated as the square root of its mean variance where . We replace with the series approximation above and take advantage of that in this approximation . In addition we write the dot product as an explicit sum, taking care to have separate indices for the terms in the square of the dot product, obtaining . The product of these two sums can be re-written as the sum of 2 terms:
Since the sub-terms in the second term above are only non-zero for, and since and are uncorrelated, and since the average of a product of an uncorrelated pair of Gaussian distributed variables is the product of their averages, the second term vanishes due to the fact that . Since the variance of a is , the final and familiar result is . In other words, uncorrelated contributions to the total uncertainty of a function of observables are to be added in quadrature.