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On the residual lifelengths of the remaining components in an \(n - k+1\) out of \(n\) system. (English) Zbl 1141.62081

Summary: Suppose that a system consists of \(n\) independent components and that the lifelength of the \(i\) th component is a random variable \(X_i\) \((i=1,2,\dots, n)\). For \(k\in \{1,2,\dots ,n - 1\}\), denote by \(X_1^{(k)},X_2^{(k)},\dots ,X_{n-k}^{(k)}\) the residual lifelengths of the remaining functioning components following the \(k\) th failure in the system. We discuss the joint distribution of these exchangeable random variables. In addition, we identify the conditions sufficient to guarantee the independence of the residual lifelengths.

MSC:

62N05 Reliability and life testing
62E15 Exact distribution theory in statistics
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