Large errors and severe conditions

Physical parameters that can assume real-number values over a continuous range are generally represented by inherently positive random variables. However, if the uncertainties in these parameters are significant (large errors), conventional means of representing and manipulating the associated variables can lead to erroneous results. Instead, all analyses involving them must be conducted in a probabilistic framework. Several issues must be considered: First, non-linear functional relations between primary and derived variables may lead to significant “error amplification” (severe conditions). Second, the commonly used normal (Gaussian) probability distribution must be replaced by a more appropriate function that avoids the occurrence of negative sampling results. Third, both primary random variables and those derived through well-defined functions must be dealt with entirely in terms of their probability distributions. Parameter “values” and “errors” should be interpreted as specific moments of these probability distributions. Fourth, there are pragmatic reasons for seeking convenient analytical formulas to approximate the “true” probability distributions of derived parameters generated by Monte Carlo simulation. This paper discusses each of these issues and illustrates the main concepts with realistic examples involving radioactivity decay and nuclear astrophysics.