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Conjunction of two probability laws can give rise to a possibility law. Using two probability densities over two disjoint ranges, we can define the fuzzy mean of a fuzzy variable with the help of means two random variables in two disjoint spaces.

Zadeh [

In this article, using the superimposition of sets, we have attempted to define the expected value of a fuzzy variable in term of expected values of two random variables in two disjoint spaces. It can be seen that the expected value of a fuzzy number is again a fuzzy set.

Let

Further, the expected value of

where the integral is absolutely convergent.

Let

where

A fuzzy set

A

The membership function of a fuzzy set is known as a possibility distribution [

number by a triad

nuous, monotone and non-decreasing, while the right reference function is left continuous, monotone and non- increasing. The above definition of a fuzzy number is known as an L-R fuzzy number.

Kandelâ€™s Definition of a Fuzzy MeasureKandel [

(1)

(2)

(3) If

(4) If

Let _{ }be a

Now

Kandelâ€™s definition of a fuzzy expected value is based on the definition of the fuzzy measure. However, the fuzzy measure being non-additive is not really a measure.

Baruah [

We propose to define the fuzzy expected value or the possibilistic mean based on the idea that two probability measures can give rise to a possibility distribution. In other words, the concerned possibilistic measure need not be fuzzy at all.

Accordingly, we propose to define a possibilistic mean as follows: Let

set

where _{ }be

where

Thus, from (2) and (3), we get the possibilistic mean of

where

Equation (4) is our required result that shows that poissibilistic mean of a fuzzy variable is again a fuzzy set.

To illustrate the result (4), we take

where

is the probability density function in

The complementary probability distribution or the survival function is given by

where

Therefore, the expected value of a uniform random variable

and similarly, the expected value of another uniform random variable

Equations (9) and (10) together give the expected value of a triangular fuzzy variable in

where

Equations (4) and (11) show that the expected value of a fuzzy number is again a fuzzy set.

The very definition of a fuzzy expected value as given by Kandel is based on the understanding that the so called fuzzy measure is not really a measure in the strict sense. The possibility distribution function is viewed as two reference functions. Using left reference function as probability distribution function and right reference function as survival function, in this article we redefine the expected value of a fuzzy number which is again a fuzzy set.