The Stochastic Properties of Conditional Spacings Based on an Unfailed Series System

In this paper, we consider the random properties of conditional spacings composed of surviving elements based on an unfailed series system. First, we obtain the marginal survival function of conditional spacings when the component lifetime distribution is an arbitrary continuous random variable. Based on this, we establish a random comparison of conditional spacings, and obtain a suffcient condition is given for the joint density function of conditional spacings is ISO^*. Then, the MTP₂ dependence of the conditional spacings and the RR₂ in pairs of the joint survival function are also obtained. In the end, the case of conditional spacings when the component lifetimes are independent Exponential is investigated, and some statistical properties of conditional spacings are naturally investigated.