1. Introduction
Zadeh [1] introduced the concept of fuzziness into the realm of mathematics. Accordingly, various authors have studied the mathematics related to the fuzzy measure and the associated fuzzy expected value [2] -[7] studied the fuzzy expected value and its associated results by defining the fuzzy expected value in terms of fuzzy measure. In their definition they tried to find the fuzzy expected value of a possibility distribution. In [8] , authors developed a new method of analysis of possibilistic portfolio that associates a probabilistic portfolio. Similar works were done in associating possibility and probability [9] [10] . In [11] [12] , the author tries to establish a link between possibility law and probability law using a concept discussed in the paper called set superimposition [13] . In [14] , the author tries to establish a link between and randomness.
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.
2. Definitions and Notations
Let
be a continuos random variable in the interval
with probability density function
and probability distribution function
. Then
![](//html.scirp.org/file/1-1680141x9.png)
Further, the expected value of
would be
(1)
where the integral is absolutely convergent.
Let
be a set and
then we can define a fuzzy subset
of
as
![](//html.scirp.org/file/1-1680141x16.png)
where
is the fuzzy membership function of the fuzzy set
for an ordinary set,
or 1.
A fuzzy set
is called normal if
for at least one
.
A
-cut
for a fuzzy set
is an ordinary set of elements such that
for
, i.e.
.
The membership function of a fuzzy set is known as a possibility distribution [15] . We usually denote a fuzzy
number by a triad
such that
and
.
, for
, is the left reference function and for
is the right reference function. The left reference function is right conti-
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.
Kandel’s Definition of a Fuzzy Measure
Kandel [5] [16] has defined a fuzzy measure as follows: Let
be a Borel field (
-algebra) of subset of the real line
. A set function
defined on
is called fuzzy measure if it has the following properties:
(1)
(
is the empty set);
(2)
;
(3) If
with
, then
;
(4) If
is a monotonic sequence, then
Clearly,
. Also, if
, then
.
is called a fuzzy measure space.
is the fuzzy measure of
.
Let
and
. The function
is called a
-measurable function, if
for all
. In their notations, fuzzy expected value is defined as follows: Let
be a
- measurable function such that
. The fuzzy expected value (FEV) of
over a set
with respect to the measure
is defined as
.
Now
is a function of the threshold
. The calculation of FEV
then consists of finding the intersection of the curves of
. The intersection of the curves will be at a value
so that FEV
as in the diagram.
![]()
3. Definition of an Expected Value of Fuzzy Number
Kandel’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 [13] has shown that instead of expressing a fuzzy measure in
, if we express the possibility distribution first as a probability distribution function in
and then as a complementary probability distribution function in
, the mathematics can be seen to be governed by the product measure on
and
. As such, the question of non-additivity of the fuzzy measure does not come into picture.
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
be a fuzzy variable in the fuzzy
set
. We divide
into two intervals
and
such that
and
. Let
be a random variable on
. Then from (1), the mean of
would be
(2)
where
is the concerned probability density function defined on
. Let the mean of the random variable an
be
(3)
where
is the concerned probability density function defined on
.
Thus, from (2) and (3), we get the possibilistic mean of
as
(4)
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
, a triangular number such that
and
. The probability distribution function is given by
(5)
where
(6)
is the probability density function in
.
The complementary probability distribution or the survival function is given by
(7)
where
and the probability density function in
is
(8)
Therefore, the expected value of a uniform random variable
on
is
(9)
and similarly, the expected value of another uniform random variable
on
is
(10)
Equations (9) and (10) together give the expected value of a triangular fuzzy variable in
as
(11)
where
.
Equations (4) and (11) show that the expected value of a fuzzy number is again a fuzzy set.
4. Conclusion
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.