Conditional residual lifetimes of coherent systems

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n$n$ independent and identically distributed (i.i.d.) components under the condition that at least j$j$ and at most k−1$k-1$ (j<k$j<k$) components have failed by time t$t$. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.