The mean wasted life time of a component of system

A reliability inspection model in which a component of a technical system has lifetime XX and inspection time SS is considered. It is assumed that XX and SS are random variables with absolutely continuous joint distribution function FX,SFX,S, and the joint probability density function fX,SfX,S. Firstly, we consider mean residual life function of the component under two different setups of inspection. Secondly, we consider an inspection model where at the inspection time the component is replaced with its spare regardless of whether the component is alive or failed at this time. Under condition that 0