Study On Some Properties Of Three Kinds Of The Weibull Process

Weibull distribution has more extensive practical application than exponentialdistribution because of it’s three parameters, and with respect to the Poisson process,Weibull process has wider application range, and more close to the real life. SunRongheng in the [1] introduces in detail the Poisson process, compound Poisson processand the filtered Poisson process and its application, but these random increments aresmooth, and in real life there are many approximately consistent with incremental notsmooth Weibull process, so the Weibull process research has certain theory value andoperation significance.The main contents of this paper include:(1)On the basis of the mathematical model of the Weibull process, compoundWeibull process and the filtration Weibull process gived in [10] to [13], this paper makesfurther study on their statistical properties and distribution properties, obtaines their meanfunction, variance function, correlation function, covariance function and thecharacteristic function, and studies the arrival time and the arrival interval’s distributionof Weibull process. And some applications of the results of this paper in real life havebeen demonstrated.(2) This paper takes special value, obtains the conclusion that Weibull process is theexceptional case of compound Weibull process, compound Weibull process is theexceptional case of filtration Weibull process. This result conforms to the conclusion thatPoisson process is the exceptional case of compound Poisson process, compound Poissonprocess is the exceptional case of filtration Poisson process.(3)This paper orderes β=1, compares the results with the Poisson process,compound Poisson process and the filtration Poisson process’ characteristics given in thethird chapter,then obtaines the conclusion that Weibull process is the promotion process of Poisson process. This not only shows the model and conclusions of this paper promotePoisson process’ relevant results, but also to some extent, validate the results of this paperis correct.