Consider a finite set of alternatives, and an associated collection of random variables representing some given relevant property of each alternative, such as its utility, cost or reliability. Suppose that there is a process that selects exactly one of these alternatives, an alternative with an extreme (maximal or minimal) value. In economics, discrete-choice analysis based on random utility theory is perhaps the most well-known modelling approach to such processes. Among practitioners, it is widely believed that in order to analyze such situations, one has to resort to particular parametric forms of the underlying probability distributions of the random variables. We here show that this is not needed. Indeed, parametric forms impose unnecessary theoretical and empirical constraints. Moreover, in the case of statistical independence, we provide necessary and sufficient conditions for the invariance property that all random variables – representing the relevant property of the alternatives – have the same distribution, conditional upon being selected.