Distributions with fuzziness and randomness

Both cognitive and noncognitive uncertainties can be present in the same variable. The non-cognitive uncertainty of a variable can be described by its own probability density function (PDF); whereas the cognitive uncertainty of a random variable can be described by the membership function for its fuzziness and its α-cuts. A PDF called fuzzy-random PDF is proposed in this paper based on considering the combined effects of both cognitive and non-cognitive uncertainties for the variable. The variable is assumed to have a fuzzy mean and a non-fuzzy standard deviation. The fuzzy-random PDF is defined as the marginal density function of the multiplication of its normalized membership function and its random distribution. Relationships for the means and variances among the fuzzy-random distribution, normalized membership function, and random distribution were developed. The moments method and discrete method were proposed for dealing with the fuzzy-random PDF