Distribution of Order Statistics for Exchangeable Random Variables

Let T-1,..,T-n be exchangeable random variables and suppose that T-i:n represents the ith order statistic among T-i's, i = 1,..., n. In this paper some expressions for the joint distribution of (T-i:n,.., T-n:n), marginal distribution of T-i:n and the joint distribution of (T-r:n, T-k:n), 1 <= r <= k <= n in terms of the joint distribution (or joint reliability) function of Ti's are provided. Using these and when {T-1,..,T-n}is a sequence of lifetimes, some expressions for the mean residual life functions of a n - k + 1-out-of-n system, H-n(k)(t) = E(T-k:n - t vertical bar T-1:n > t) and M-n(r,k) (t) = E(T-k:n - t vertical bar T-r:n > t) ,1 <= r <= k <= n in terms of the joint survival function of T-i's are given. Also, some examples are provided.