A note on the conditional residual lifetime of a coherent system under double monitoring

In this note, we derive some mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at time t1 the jth failures has occurred and at time t2 the kth failures (j < k) have not occurred yet. Based on the mixture representations, we then discuss the stochastic comparisons of the conditional residual lifetimes of two coherent systems with i.i.d. components.