The probability distribution and fuzzy system based on bounded product implication

In this paper, a method of constructing joint probability density function and fuzzy system using non-singleton fuzzifier is proposed based on a set of input-output data. Firstly, we create fuzzy relations using bounded product implication operator. With the fuzzy relations, we get the joint probability density function of binary random variable. Secondly, the marginal density functions and numerical characteristics of bounded product distribution are discussed separately. Finally, the regression function is obtained based on the probability distribution. It is actually the center of gravity fuzzy system, which further reveals the probability sense of fuzzy systems with center of gravity method. Meanwhile, the sufficient condition for the fuzzy system as an universal approximator is given.