Distribution of Order Statistics for Exchangeable Random Vari-ables

Let T1; : : : ; Tn be exchangeable random variables and suppose that Ti:n rep- resents the ith order statistic among Ti’s, i = 1; : : : ; n. In this paper some expressions for the joint distribution of (T1:n; : : : ; Tn:n), marginal distribution of Ti:n and the joint distribution of (Tr:n; Tk:n), 1  r  k  n in terms of the joint distribution (or joint relia- bility) function of Ti’s are provided. Using these and when fT1; : : : ; Tng is a sequence of lifetimes, some expressions for the mean residual life functions of a n 􀀀 k + 1-out-of-n system, Hkn (t) = E(Tk:n 􀀀 tjT1:n > t) andMr;k n (t) = E(Tk:n 􀀀 tjTr:n > t), 1 r  k  n in terms of the joint survival function of Ti’s are given. Also, some examples are provided.