2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222136For independent $X$ and $Y$ in the inequality $P(X\leq Y+μ)$, we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).Published in at http://dx.doi.org/10.1214/08-AAP536 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability62G32, 60E15 (Primary) 92C55. (Secondary)text