Chance Measure And Its Applications

Fuzziness and randomness are two kinds of basic uncertainties in real world, andcredibility theory and probability theory are two branches of mathematics for dealingwith fuzzy phenomena and random phenomena, respectively. As the improvement ofunderstanding about uncertain phenomena, some researchers begin to study the com-plex system which includes both fuzziness and randomness. In 1978, Kwakernaak firstrealized this problem and introduced a fuzzy random variable, and Liu proposed a ran-dom fuzzy variable in 2002. In nature, fuzzy random variable is a random elementtaking values in the set of fuzzy variables, and random fuzzy variable is a fuzzy ele-ment taking values in the set of random variables. However, fuzziness and randomnessoccur simultaneously in real problems. Hence, Liu proposed the concepts of chancespace and hybrid variable for describing this phenomena.For dealing with hybrid variable, a new measure on chance space is necessary.This dissertation defines a chance measure on the basis of credibility measure andprobability measure, and proves the consistency theorem. That is, if chance spacedegenerates to a credibility space, then the chance measure degenerates to the credi-bility measure, and if chance space degenerates to a probability space, then the chancemeasure degenerates to the corresponding probability measure. Furthermore, as appli-cations of chance measure, some important concepts about hybrid variable are studiedsuch as chance distribution, critical value, expected value, variance, moments, distance,convergence and so on. Then an elementary mathematical theory for dealing with fuzzi-ness and randomness simultaneously is founded, which will be called chance theory.In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects: (1) a new concept of chance measure on chance space isdefined; (2) the expected value of hybrid variables is defined as an application of chancemeasure; (3) the chance distribution, critical value, distance and convergence conceptsof hybrid variables are discussed.