A Summary on Fuzzy Probability Theory

Fuzzy probability theory is an extension of probability theory to dealing with mixed probabilistic/non-probabilistic uncertainty. It provides a theoretical basis to model uncertainty which is only partly characterized by randomness and defies a pure probabilistic modeling with certainty due to a lack of trustworthiness or precision of the data or a lack of pertinent information. The fuzzy probabilistic model is settled between the probabilistic model and non-probabilistic uncertainty models. The significance of fuzzy probability theory lies in the treatment of the elements of a population not as crisp quantities but as set-valued quantities or granules in an imprecise manner, which largely complies with reality in most everyday situations. Probabilistic and non-probabilistic uncertainty and imprecision can so be transferred adequately and separately to the results of a subsequent analysis. This enables best case and worst case estimates in terms of probability taking account of variations within the inherent non-probabilistic uncertainty. The development of fuzzy probability theory was initiated by H. Kwakernaak with the introduction of fuzzy random variables in 1978. The usefulness of the theory has been underlined with various applications beyond mathematics and computer science. An increasing interest in fuzzy probabilities and related concepts has been developed, in particular, in engineering. In this summary, a general introduction to fuzzy probability theory is given. Detailed mathematical descriptions and discussions in an engineering context are provided in.