The Depicition Of Multivariate Lifetime Distribution Based On Conditional Failure Rate

In reliability theory, failure rate function is a kind of quite importa -nt characteristic function, because it shows to be failure probability in condition that it will be regular until it operates at moment t.So it is,f-ailure rate function can decide failure intensity at each moment and be instantaneous characteristic quantity of reliability.From this, failure ta-te function can be thought to have visual meanings and be easily obtained. Besides, failure rate function can only define one-dimensional lifetime distribution.However, for two-dimensional joint lifetime distribution consisting of two dependent components, it is difficult to depict it by only failure rate function. Because the relation of failure rate and conditional failure rate is similar to the relation of distribution and conditional distribution, condition failure rate function has visual meanings and is e-asily obtained. But for three-dimensional and even n-dimensional case ,the depiction is quite difficult. First, according to the definition of n-dimensional density function, this thesis decomposes it into the conditional probability about variable alone, which can be expressed by conditional failure rate function about variable alone. We obtains three kinds of conditional failure function throught these conditional probability probabilities and depicts three-dimensional lifetime distribtution a-nd even to extend the depiction in n-dimensional lifetime distribution. Thus, we obtain a effective and new way to depict multivariate lifetime distribution. Meanwhile, at the same reason, the thesis depicts three-dim -ensional jont failure functions and extend the depiction in n-dimensio -nal joint failure rate function.According to the relation of conditional failure rate function and Multivariate lifetime distribution built, we discuss some important qualities related to conditional failure rate function and apply the depiction to some examples.Exponential distribution is the most important distribution in relia-bility theory.In reliability history.famous Marshall-Olkin applied the fatal shock model with three mutually independent shock sources to build bivariate exponential distribution. But, many documentaries have all mentioned that it has been quite difficult to depict three-dimension -al and even n-dimensional dependent distribution. Therefrom, this thesis puts forward the shock model with seven individually independent sources and structure three-dimensional exponential distribution, which its edge distribution is only bivariate distribution of Marshall-Olkin. S-eeing that the analysis process is loaded down with trivial details an$ is difficulty extended in multivariate case.However,taking advantage of relation between multivariate failure rate and distribution density function built by the author and endowing conditional failure rates with reasonable value joint with the fatal shock meaning, we easily obtain three-dimensional and even n-dimensional exponential distribution.Finally,the thesis obtains some theorems about three-dimensional Exponential distribution. Throught the multivariate exponential distribution model built, it even reflects the advantage of depiction dependence.