Characterizing random variables in the context of signal detection theory

By generalizing properties of the standard normal model of signal detection theory, we are able to characterize symmetric, partially stable random variables in terms of a pair of observable relations coupling yes-no and forced-choice paradigms. One of these relations is provided by the 'area theorem': the area subtended by a receiver operating characteristics equals the probability of a correct response in the corresponding forced-choice task. The other relation involves the algebraic notion of bisymmetry. It turns out that most partially stable random variables are actually stable. Accordingly we can interpret our results as characterizing detection models in which 'signal' and 'noise' random variables arise as sums of many small independent contributions.