## 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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### References:

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