The number of failed components in a coherent working system when the lifetimes are discretely distributed

TLDR: The probability that there are exactly i, $$i=0,\ldots ,n-k,$$ failures in a k-out-of-n system under the condition that it is operating at time t, is computed.

Abstract: In this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.