Statistica Sinica

James C. Fu

Abstract:Reliability of a large linearly connected engineering system is closely associated with the probability of a certain pattern occurring in a sequence of coin tossing. In this paper a new method is developed to show that if the failure probabilities of components are very small then the reliability of the system can be approximated by a Poisson random variable. The proof of the result is essentially dependent on the Markov property of coin tossing. It is more direct and elementary than the standard tools such as Bonferroni inequalities and the Stein-Chen method. The result is also extended for the cas e that the failure probabilities of components are different. Necessary and sufficient conditions for Poisson convergence are also obtained. Numerical upper and lower bounds of reliabilities of linearly connected systems developed from the new method are obtained and compared with bounds derived from the Stein-Chen method.

Key words and phrases:Reliability, linearly connected system, coin tossing, pattern, Poisson convergence.