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Various concepts of stochastic comparison between random variables have been defined and studied in the literature since they are useful in reliability modeling and in economics applications and as mathematical tools for proving important results in applied probability. Some well-known orders that have been introduced and studied in reliability theory such as the usual stochastic order, the hazard rate order, mean residual life order, mean inactivity time order, were mentioned in this dissertation. As for stochastic orders, several non-parametric classes of life distributions were proposed in this dissertation to provide some characterizations of inactivity time. First, we studied mean inactivity time, and provided the properties of it in reliability theory, and provided the relation between it and usual stochastic order, hazard rate order, reversed hazard rate order, total time on test transform order. Second, we studied new non-parametric class of life distributions based on the median of inactivity time and studied its reliability properties. Some new results of the proposed class were given including some closure properties and characterizations.
We also studied a new stochastic ordering based on the median of inactivity time and revealed its relationship with other well-known orders. Also, we provided some characterizations of some well-known life distributions via their median inactivity time functions. We studied further new non-parametric class of life distributions based on the variance of inactivity time. Also, we studied closure properties of this class under relevant reliability operations such as mixing, convolution and formation of coherent systems. We showed that variance inactivity time is closed under convolution, mixing, and coherent systems; The applications of inactivity time functions were presented in this dissertation, where we studied the comparison between mean inactivity time class and median inactivity time of some distributions. Connection between mean inactivity time and variance inactivity time was presented. The problems of testing increasing mean inactivity time, median inactivity time, and increasing variance inactivity time were investigated. Finally, we presented the conclusion and further extensions.
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The aim of this dissertation is to address the two most interesting concepts in this area and those are stochastic orderings and ageing notions then provide several practical applications of these new notions to areas such as medicine, engineering, manufacturing, economics and to the problem of biological ageing.
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1. Studying related published literature.
2. Identifying the required mathematical and simulation tools.
3. Developing and analyzing the proposed schemes.
4. Developing the simulation work.
5. Evaluting the efficiency and importance of the proposed schemes.
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