Computing moments of discrete order statistics from non-identical distributions

In life-testing and reliability studies, it is commonly assumed that the lifetimes come from a continuous distribution. While this often makes derivations more tractable, under certain circumstances, it may be more realistic to assume a discrete distribution. Recent literature has heavily focused on the former and with this, results have been aimed at deriving properties in the case of independent and identically distributed random variables and associated order statistics. In this paper, we consider the case of discrete order statistics from non-identical distributions. We derive expression for single and product moments, and with this, covariances. We demonstrate the results for the geometric and binomial distributions.