We propose a random quasi-linear utility model (RQUM) where quasi-linear utility functions are drawn randomly via some probability distribution π, and utility ties are broken by a convenient lexicographic rule. We characterize RQUM and identify π uniquely in terms of stochastic choice data. McFadden’s (1973) additive random utility model is obtained as a special case where utility ties have a zero probability in all menus. Another distinct case of RQUM captures finite populations and derives π with a finite support. Our main axioms are testable. They prohibit context and reference dependence, and also modify the non-negativity of Block-Marschack polynomials for monetary cost variations. We also characterize RQUM through a stronger version of McFadden and Richter’s (1990) axiom of revealed stochastic preferences (ARSP). This approach extends to incomplete datasets.