If the random variable TT denotes the lifetime of a unit, then the random variable T(t)=[t−T∣T≤t]T(t)=[t−T∣T≤t], for a fixed t>0t>0, is known as the past lifetime. In this study, we present some new properties of the mean and variance for past lifetime classes (orderings). In addition, we consider an (n−r+1)(n−r+1)-out-of-nn system with identical components where it is assumed that the lifetimes of the components are i.i.d. We assume that the system fails before time xx,x>0x>0. Under these conditions, we are interested in studying the variance time elapsed since the failure of the components. Several properties of this function are studied and an example is provided. Finally, some applications in economic theory are described with real data.