Random Uncertainty Distribution of Measurements of a Complex Quantity withPartially-Correlated Real and Imaginary Parts (or Any Pair of Partially-Correlated Variables)

摘要

The report presents a rigorous derivation, from first principles, of the randomuncertainty distribution of a sample of measurements of a complex number, allowing for partial linear correlation between the complex number's real and imaginary parts, and assuming that the underlying law of joint distribution of the real and imaginary parts is the law of Gaussian normal correlation. This is the equivalent in two dimensions of the Student's t distribution in one dimension. The principal result is an expression for the probability that the underlying true value of the complex number is in the neighborhood of any particular point in the complex plane, in terms of the means of the sample's real and imaginary parts, their standard deviations and their correlation coefficient. This probability density distribution is such that the contours of equal probability density are ellipses centered on the sample's two-dimensional mean, and a simple closed-form expression is given for the probability that the true value is inside any particular ellipse, which relates directly to the customary expression by confidence levels of uncertainties in measurement. (Copyright (c) Crown Copyright 1995.)