Fuzzy improved distribution function and order statistics

A new concept of fuzzy improved distribution function based on fuzzy ordering defined on the support set of a given random variable is introduced. A fuzzy improved random variable based on this distribution function is defined. The existence of such a fuzzy improved random variable follows from probability integral transformation. The order statistics based on the fuzzy ordering and fuzzy improved distribution function are considered and their distributions are studied. The need for fuzzy improved order statistics arises in reliability analysis of sensitive systems where the lifetimes of the components and the system are considered by taking into account some degree of smallness defined by membership function of a fuzzy set. Examples and graphical representations of fuzzy improved distribution functions are provided.