Consider a system consisting of ‎n‎‎ ‎independent binary ‎components. ‎Suppose ‎that ‎each component has a random weight and the system works, at time ‏‎t, ‎if ‎the ‎sum ‎of ‎the ‎weight ‎of all ‎working ‎components ‎at ‎time ‎‎t‎‎, ‎is above ‎a pre-specified value k.‎ We ‎call ‎such a‎ ‎system ‎as ‎random-‎weighted-‎k‎‎-out-of-‎n‎‎ ‎system. ‎In ‎this ‎paper, we investigate the effect of the component weights and reliabilities on the system performance and show that the larger weights and reliabilities, the larger lifetime (with respect to the usual stochastic order). ‎We ‎also ‎show ‎that ‎the ‎best ‎‎random-‎weighted-‎k‎‎-out-of-‎n‎‎ ‎system ‎is ‎obtaind ‎when ‎the components with the ‎more ‎weights ‎have simultaneously ‎more ‎reliability. The reliability function and mean time to failure of a ‎random-‎weighted-‎k‎-out-of-‎n‎ ‎system are stated based on the reliability function of coherent systems. Furthermore, a simulation algorithm is presented to observe the mean time to failure of ‎random-‎weighted-‎k‎‎-out-of-‎n‎ ‎system.

In this paper, the worst allocation of deductibles  and limits in layer policies are discussed from the viewpoint  of the insurer. It is shown that if n independent and identically distributed exponential risks are covered by the layer policies and  the policy limits are equal, then the worst allocation of deductibles from the viewpoint of the insurer is (d‎, ‎0‎, ‎..., ‎0)‎.