{"product_id":"the-mathematics-of-fuzziness-realities-and-beyond-von-hemanta-baruah","title":"THE MATHEMATICS OF FUZZINESS: REALITIES AND BEYOND","description":"\u003cp\u003eImposing 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.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783838399416\"\u003e\u003ch3\u003eRandomness - Fuzziness Consistency Principle\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783838399416","offer_id":39497045803101,"sku":"9783838399416","price":49.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/c378120c-2722-4f91-90fc-3c6f0480f97d.jpg?v=1757655722","url":"https:\/\/shop.autorenwelt.de\/products\/the-mathematics-of-fuzziness-realities-and-beyond-von-hemanta-baruah","provider":"Autorenwelt Shop","version":"1.0","type":"link"}