We consider a nonnegative random variable T representing the lifetime of a system. We discuss the residual lifetime $T_X=(T-X|T \gt X)$, where X denotes the random age of the system. We also discuss the mean residual life (MRL) of T at the random time X. It is shown that the MRL at random age (MRLR) is closely related to some well-known variability measures. In particular, we show that the MRLR can be considered a generalization of Gini’s mean difference (GMD). Under the proportional hazards model, we show that the MRLR gives the extended GMD and the extended cumulative residual entropy as special cases. Then, we provide a decomposition result indicating that the MRLR has a covariance representation. Some comparison results are also established for the MRLs of two systems at random ages.

In this short note, we show that every convex, order-bounded above functional on a Fréchet lattice is automatically continuous. This improves a result in Ruszczyński and Shapiro ((2006) Mathematics of Operations Research 31(3), 433–452.) and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in Krätschmer et al. ((2017) Finance and Stochastics 18(2), 271–295.).