Location and scale fuzzy random variables

Several interpretations have been proposed so far for fuzzy random variables to describe imprecision and randomness. In this paper, a novel notion of fuzzy random variable was proposed to model fuzziness and randomness in a statistical procedure. For this purpose, a frequently used family of probability distributions called location and scale were first employed as the origin of randomness. Then, α-values of fuzzy numbers were combined with randomness to describe the fuzziness in the nature of the processes to produce a new concept of fuzzy random variable called location and scale fuzzy random variables. Then, some essential statistical features of the proposed fuzzy random variables including fuzzy cumulative distribution function, fuzzy expectation, exact variance and imprecise probability of an interval were discussed. The classical method of moment estimator was also developed to estimate the location and scale parameters. The developed technique was illustrated via several numerical evaluations. As an real-life application of the proposed fuzzy random variable, the reliability functions of k-out-of-n system and some reliability evaluation criteria were introduced and interpreted. Some numerical examples were also presented to illustrate the calculation of the proposed fuzzy system reliability criteria.