Reliability properties of some statistical distributions

Abstract

Reliability, as a human attribute, has been praised for a very long time. It emerged with a technological meaning just after World War I and was then used in connection with comparing operational safety of one-, two-, and four-engine airplanes. The reliability was measured as the number of accidents per hour of and#64258;ight time. Toward the end of the 1950s and the beginning of the 1960s, interest in the United States was concentrated on intercontinental ballistic missiles and space research, especially connected to the Mercury and Gemini programs. An association for engineers working with reliability questions was soon established. In the 1970s interest increased, in the United States as well as in other parts of the world, in risk and safety aspects connected to the building and operation of nuclear power plants. A detailed history of reliability technology is presented, by Rausand and Hoylandand (2004) and others. In the majority of industries a lot of eand#64256;ort is presently put on the analysis of risk and reliability problems. In Reliability, Biological Sciences, Forensic Science, and related and#64257;elds, comparison of two diand#64256;erent products of two diand#64256;erent brands is needed. In Reliability and Survival Analysis, the remaining lifetimes of a component at diand#64256;erent time of its life span need to be compared to determine whether the component is aging with time. In this context, one can have questions in mind as to and#8722;What do people mean when they say that one product (say, A) is better than another (say, B)? How do we compare two products, say, A and B, the lifetimes of which are random variables, say, X and Y respectively? The simplest way of comparing the two products is by the comparison of their means, which is based on only two numbers measuring only the centers of the distributions, and is often not very informative. In addition, the means sometimes do not exist. One can compare two random variables having the same mean in terms of measures of dispersion. Again, such a comparison is based on only two numbers having the same ty