Bayesian analysis of P[Y < X] in its limiting form

The present study deals with the Bayesian estimation of the remodeled stress strength
system reliability, P[Y < X]. Here, X and Y represent strength and
stress variables respectively.The random variables X and Y have been respectively
redefined as U = min(X_1,X_2, \ldots ,X_m) and V = max(Y_1, Y_2, \ldots , Y_n) to
conceptualize the concept of limiting stress-strength reliability, P[V < U], for
meeting the requirements of the systems in defense. For such systems, the designer
wishes to attach high probability to the event that the system remains
operable at its minimum strength encountering with the maximum stress.