The concept of mean inactivity time plays a crucial role in reliability, risk theory and life testing. In this regard, we introduce a weighted mean inactivity time function by considering a non-negative weight function. Based on this function, we provide expressions for the variance of transformed random variable and the weighted generalized cumulative entropy. The latter concept is an important measure of uncertainty which is shift-dependent and is of interest in certain applied contexts, such as reliability or mathematical neurobiology. Moreover, based on the comparison of mean inactivity times of a certain function of two lifetime random variables, we introduce and study a new stochastic order in terms of the weighted mean inactivity time function. Several characterizations and preservation properties of the new order under shock models, random maxima and renewal theory are discussed.

翻译：中不活动时间的概念在可靠性、风险理论和生命测试方面起着关键作用。在这方面,我们通过考虑非负加权权重功能引入了加权中不活动时间功能。基于此功能,我们提供了变异随机变量和加权普遍累积的倍数差异的表达方式。后一种概念是不确定性的一个重要度量,取决于变化,并对某些应用环境,如可靠性或数学神经生物学感兴趣。此外,根据对两个寿命周期随机变量某种函数的中不活动时间的比较,我们采用并研究一种按加权中不活动时间函数计算的新的随机顺序。讨论了冲击模型下新顺序的若干特征和保存特性、随机峰值和更新理论。