This thesis is an extension (and generalization) of the work(s) done by Ebrahim Amini-Seresht and Baha-Eldin Khaledi( [1]), Yan Shen and Zhisheng Ye( [21]). In reliability engineering, coherent systems have a key role in the study of the reliability of technical systems. In this thesis we introduce lifetime measures like the reliability function, the hazar rate function, the reversed hazard rate function, the mean residual life time, the mean path life time. Suppose that X 1 ; :::X n be a random sample from a distribution function F that denote lifetimes of n components of a coherent system and we know the system fails at X k : n , since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining n ?k components, denoted by X ( k ) 1 ; :::;X ( k ) n ?? k are no longer independent but exchangeable. On the other hand in this study multivariate stochastic comparisons of two vectors of lifetimes of the remaining components in the two sample problems are studied and some sufficient conditions under which multivariate mixture models are compared stochastically with respect to the multivariate likelihood ratio ordering, the multivariate hazard rate ordering and the multivariate reversed hazard rate ordering are provided. These comparisons are done for different choices of the mixed distributions as well as mixing distributions. The new results obtained are applied to compare multivariate mixtures of location models. Residual life is an important concept in many disciplines, with respect to the subject investigates properties of the percentile residual life (PRL) function for a single component and for a k-out-of-n system. The shape of the component PRL function can be determined by the component failure rate function. The intimate relations between these two functions are studied first. Then we generalize the results to a k-out-of-n system by assuming independent and identical components. We show that the behavior of the PRL for a k-out-of-n system is quite different from the component PRL. We also find that for series and parallel systems, the location of the change point of the PRL is monotone in the number of components in a system. Residual life, also called remaining useful life, is an important concept in many disciplines. In survival analysis, we may evaluate the efficacy of a new drug in terms of patients’ residual lives. In reliability engineering, the information of a working machine’s residual life is needed for a future maintenance plan. From the point of view of probability theory, the residual life is a conditional random variable referring to the remaining lifetime of an item given that this item has survived up to a period of time. As a random variable, the residual life is characterized by its percentile called percentile residual life (PRL). A study on the behavior of the PRL helps to get a better understanding of the residual life.