Absolute Bounds on the Mean of Sum , Product , etc . : A Probabilistic Extension of

We extend the main formulas of interval arithmetic x1 x2 for diierent arithmetic operations x1 x2 to the case when, for each input xi, in addition to the interval xi = x i ; xi ] of possible values, we also know its mean Ei (or an interval Ei of possible values of the mean), and we want to nd the corresponding bounds for x1 x2 and its mean. A practically important class of statistical problems is related to data processing (indirect measurements). Some physical quantities y { such as the distance to a star or the amount of oil in a given well { are impossible or diicult to measure directly. To estimate these quantities, we use indirect measurements, i.e., we measure some easier-to-measure quantities x 1 ; : : : ; x n which are related to y by a known relation y = f(x 1 ; : : : ; x n), and then use the measurement results e x i (1 i n) to compute an estimate e y for y as e