On the residual lifelengths of the remaining components in an n−k+1 out of n system

Suppose that a system consists of n independent components and that the lifelength of the ith component is a random variable . For k[set membership, variant]{1,2,...,n-1}, denote by , the residual lifelengths of the remaining functioning components following the kth 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.

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