THE MATHEMATICS OF FUZZINESS: REALITIES AND BEYOND

Imposing a probability law on an interval in which a normal fuzzy number has been defined, and then trying to find consistency between randomness and fuzziness is not logical. If we need to establish a probability law followed by a random variable defined in a given interval, there are mathematical formalisms in the theory of statistical inferences to do so. Instead, defining a normal fuzzy number around a point, and then using a conversion factor to deduce a probability density function from the fuzzy membership function is against statistical norms. Probability densities are not found in this way. In fact, the left reference function of a normal fuzzy number is a probability distribution function, and the right reference function is a complementary probability distribution function. Hence, we need two probability laws to define the membership function of a normal fuzzy number. That the membership function is expressible as a distribution function and a complementary distribution function on the left and on the right respectively of the value with unit membership should be the real randomness- fuzziness consistency principle.

Autorentext

Hemanta K. Baruah, Ph.D.(Maths.,I.I.T.,Kharagpur), Professor of Statistics, Gauhati University, India, has been currently working on an invitational research project on Uncertainty Analysis of Atmospheric Dispersions, sponsored by the Department of Atomic Energy, Government of India. His research interests are in Graph Theory and Fuzzy Mathematics.