Joint Distribution of a Random Sample and an Order Statistic: a New Approach With an Application in Reliability Analysis

Abstract

This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an (n - r + 1)-out-of-n system, which is intact until at least n - r + 1 of the components are alive, and it fails if the number of failed components exceeds r. The (n - r + 1)-out-of-n system is a preferred type of redundancy of fault-tolerant systems used in reliability engineering and finds wide applications in industrial and military systems. Well-known examples of this system are multiengine systems in airplane, multipump systems in hydraulic control, multicylinder engines in automobile, the radar systems, communication systems with multiple transmitters, a web host with several servers. The lifetime of the system is the rth order statistic. Assuming that some of the components failed but the system is still functioning, using the results presented in this paper it is possible to find an expected value of the number of inspections we need to do for detecting certain number of failed components.