CONFIDENCE INTERVALS FOR INDEPENDENT EXPONENTIAL SERIES SYSTEMS

Abstract Suppose X1, X2, ···, Xn are independent identically distributed exponential random variables with parameter λ1. Let Y1, Y2, ···, Ym also be independent identically distributed exponential random variables with parameter λ2, and assume that X's and Y's are independent. The problem is to estimate R(t) = e−(λ1+λ2)t. A procedure for determining on exact (1 − α) level lower confidence bound for R(t) is presented. let U = min(Σn t = 1Xi, Σm i = 1Yi), and K = {largest i ≤ n: Σj i − 1Xi ≤ U} + {Largest i ≤ m: Σj i = 1 Yi ≤ U}. Then given K = k, it is shown that U has a gamma distribution with parameters k and λ1 + λ2. Hence, a lower confidence bound for R(t) can be obtained. The suggested procedure is then compared with others presented in the literature.