Some properties of order statistics when the sample size is random

The study of r-out-of-n systems is of utmost importance in reliability theory. In this note, we study closure of different partial orders under the formation of r-out-of-N and (N-s)-out-of-N systems when the number of components N, forming the system, is a random variable having support {k, k+1, ...}, where k is a fixed positive integer, r is an element of {1, ...., k} and s is an element of {0, 1, ...., k - 1}. This generalizes quite a few results already known in the literature. We also study the closure of different partial orders when two systems are formed out of different random number of components.