ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.9(2010) No.4,pp.486-492
A Note of the Expected Value and Variance of Fuzzy Variables
Zhigang Wang1 ∗ , Fanji Tian2
1
2
Department of Applied Mathematics, Hainan University,Haikou,570228,China
Institute of Mathematics and Computer Science,Hubei University,Wuhan,430062,China
(Received 15 May 2009 , accepted 9 December 2009)
Abstract: The emphasis in this paper is mainly on some properties expected value operator and variance of
fuzzy variables,the expceted value and variance formulas of three common types of fuzzy variables.In the
end, the expceted value and variance of geometric Liu process are given.
Keywords: fuzzy variables;expected value ;variance; geometric Liu process.
1
Introduction
Probability theory is a branch of mathematics for studying the behavior of random phenomena based on the normalality,nonnegativity and countable additivity axioms. however,the additivity axiom of classical measure theory has been
challenged by many mathematicians.Sugeno[1] generalized classical measure theory to fuzzy measure theory by replacing additivity axiom with monotonicity and semicontinuty axioms.In order to deal with general uncertainty,self-duality
and countable subadditivity are much more important than continuity and semicontinuity. For this reason ,Liu[2] founded
an uncertainty theory that is a branch of mathematics based on normality ,monotonicity,self-duality and countable subadditivity axioms. Uncertainty provides the commonness of probability theory,credibility theory and chance theory.In
fact,probability,credibility,chance,and uncertain measures meet the normality,monotonicity and self-duality axioms.The
essential difference among those measures is how to determine the measure of union.Since additivity and maximality are
special cases of subadditivity,probability and credibility are special cases of chance measure,and three of them are in the
category of uncertain measure.This fact also implies that random variable and fuzzy variable are special cases of hybrid
variables,and three of them are instances of uncertain variables. The emphasis in this paper is mainly on some properties
expected value operator and variance of fuzzy variables,the expceted value and variance formulas of three common types
of fuzzy variables.In the end, the expceted value and variance of geometric Liu process are given.
2
Text
2.1 Credibility measure
In this section,we review some preliminiary concepts within the framework of credibility theory.([3-6])
Definition 1 (Liu[2]) Let Θ be a nonempty set,and 𝒫(Θ) the power set of Θ.A set function 𝐶𝑟(.)called a credibility
measure if it satisfies:
(1)(Normality) 𝐶𝑟(.) = 1;
(2)(Monotonicity) 𝐶𝑟{𝐴} ≤ 𝐶𝑟{𝐵} whenever 𝐴 ⊂ 𝐵;
(3)(Self-Duality) 𝐶𝑟{𝐴} + 𝐶𝑟{𝐴𝑐 } = 1 for any event 𝐴;
(4)(Maximality) 𝐶𝑟{∪𝑖 𝐴𝑖 } = sup𝑖 𝐶𝑟{𝐴𝑖 }for any eventa {𝐴𝑖 }with sup𝑖 𝐶𝑟{𝐴𝑖 } < 0.5.
The triplet (Θ, 𝒫(Θ), 𝐶𝑟) is called a credibility space.
The law of contradiction tells us that a proposition cannot be both true and false at the same time, and the law of
excluded middle tells us that a proposition is either true or false. Self-duality is in fact a generalization of the law of
∗ Corresponding
author.
E-mail address: [email protected]
c
Copyright⃝World
Academic Press, World Academic Union
IJNS.2010.06.30/377
Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables
487
contradiction and law of excluded middle. In other words, a mathematical system without self-duality assumption will be
inconsistent with the laws. This is the main reason why self-duality axiom is assumed.
Definition 2 (Liu[2]) A fuzzy variable is a function from a credibility space (Θ, 𝒫,𝐶𝑟) to the set of real numbers.
Definition 3 (Liu[2]) Let 𝜉 be a fuzzy variable defined on the credibility space (Θ, 𝒫,𝐶𝑟) . Then its membership
function is derived from the credibility measure by
𝜇(𝑥) = (2𝐶𝑟{𝜉 = 𝑥}) ∧ 1, 𝑥 ∈ ℛ
(1)
In practice, a fuzzy variable may be specified by a membership function. In this case, we need a formula to calculate the
credibility value of some fuzzy event. The credibility inversion theorem provides this[2] .
For fuzzy variables, there are many ways to define an expected value operator. See, for example,Dubois and Prade
[7], Heilpern [8], Campos and Gonzalez [9], Gonzalez [10] and Yager [11]. The most general definition of expected value
operator of fuzzy variable was given by Liu and Liu [3].This definition is applicable to both continuous and discrete fuzzy
variables.
2.2 Expected valueof fuzzy variables
Definition 4 (Liu and Liu [4]) Let 𝜉 be a fuzzy variable. Then the expected value of 𝜉 is defined by
∫ +∞
∫ 0
𝐸[𝜉] =
𝐶𝑟{𝜉 ≥ 𝑟}𝑑𝑟 −
𝐶𝑟{𝜉 ≤ 𝑟}𝑑𝑟
0
−∞
(2)
provided that at least one of the two integrals is finite.
The equipossible fuzzy variable on [a, b] has an expected value (𝑎+𝑏)
2 . The triangular fuzzy variable (a,b, c) has an
(𝑎+2𝑏+𝑐)
expected value
; The trapezoidal fuzzy variable (a, b, c,d) has an expected value (𝑎+𝑏+𝑐+𝑑)
.
4
4
Let 𝜉 be a continuous nonnegative fuzzy variable with membership function 𝜇. If 𝜇 is decreasing on [0,∞), then
𝐶𝑟{𝜉 ≥ 𝑥} = 𝜇(𝑥)
2 for any 𝑥 > 0, and
∫
1 ∞
𝐸[𝜉] =
𝜇(𝑥)𝑑𝑥
(3)
2 0
Example 1 A fuzzy variable 𝜉 is called exponentially distributed if it has an exponential membership function
𝜋𝑥
𝜇(𝑥) = 2(1 + exp( √ ))−1 , 𝑥 ≥ 0, 𝑚 > 0
6𝑚
(4)
√
then the expected value is 6𝑚𝜋 ln 2 .
Theorem 1 Let 𝜉 be a continuous fuzzy variable with membership function 𝜇. If its expected value exists, and there
is a point 𝑥0 such that 𝜇(𝑥) is increasing on (−∞, 𝑥0 ) and decreasing on (𝑥0 , +∞), then
𝐸[𝜉] = 𝑥0 +
Proof
1
2
∫
+∞
𝑥0
𝜇(𝑥)𝑑𝑥 −
1
2
∫
𝑥0
−∞
𝜇(𝑥)𝑑𝑥
If 𝑥0 ≥ 0,then
{
𝐶𝑟{𝜉 ≥ 𝑟} =
1
2 [1 + 1
1
2 𝜇(𝑥).
− 𝜇(𝑥)], 𝑖𝑓
𝑖𝑓
0 ≤ 𝑟 ≤ 𝑥0 ,
𝑟 > 𝑥0 .
and 𝐶𝑟{𝜉 ≤ 𝑟} = 12 𝜇(𝑥),
𝐸[𝜉]
=
=
∫ +∞
∫0
[1 − 12 𝜇(𝑥)]𝑑𝑥 + 𝑥0 12 𝜇(𝑥)𝑑𝑥 − −∞ 12 𝜇(𝑥)𝑑𝑥
∫ +∞
∫ 𝑥0
𝑥0 + 12 𝑥0 𝜇(𝑥)𝑑𝑥 − 12 −∞
𝜇(𝑥)𝑑𝑥
∫ 𝑥0
0
If 𝑥0 < 0,a similar way may prove the equation (5). The theorem 1 is proved.
Especially,let 𝜇(𝑥) be symmetric function on the line 𝑥 = 𝑥0 ,then 𝐸[𝜉] = 𝑥0 .
Example 2 Let 𝜉 be triangular fuzzy variable (a,b, c) ,then it has an expected value
𝐸[𝜉] = 𝑏 +
1
2
∫
𝑏
𝑐
𝑥−𝑐
1
𝑑𝑥 −
𝑏−𝑐
2
∫
𝑏
𝑎
𝑥−𝑎
𝑐−𝑏 𝑎−𝑏
𝑎 + 2𝑏 + 𝑐
𝑑𝑥 = 𝑏 +
+
=
𝑏−𝑎
4
4
4
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(5)
488
International Journal of NonlinearScience,Vol.9(2010),No.4,pp. 486-492
Example 3 Let 𝜉 be equipossible fuzzy variable on [a, b],then it has an expected value
𝑎+𝑏 1
𝐸[𝜉] =
+
2
2
∫
𝑏
𝑎+𝑏
2
1
1𝑑𝑥 −
2
∫
𝑎
𝑎+𝑏
2
𝑎+𝑏
2
1𝑑𝑥 =
Example 4 Let 𝜉 be trapezoidal fuzzy variable (a, b, c,d) ,then it has an expected value
𝑏+𝑐 1
𝐸[𝜉] =
+
2
2
∫
𝑐
𝑏+𝑐
2
1
1𝑑𝑥 +
2
∫
𝑑
𝑐
𝑥−𝑑
1
𝑑𝑥 −
𝑐−𝑑
2
∫
𝑏
𝑎
𝑥−𝑎 1
−
𝑏−𝑎
2
∫
𝑏+𝑐
2
𝑏
1𝑑𝑥 =
𝑎+𝑏+𝑐+𝑑
4
Especially,let 𝜉 be triangular fuzzy variable (a,b, c) and 𝑏 − 𝑎 = 𝑐 − 𝑏,then it has an expected value 𝐸[𝜉] = 𝑏.
Let 𝜉 be trapezoidal fuzzy variable (a, b, c,d) and 𝑏 − 𝑎 = 𝑑 − 𝑐,then it has an expected value 𝐸[𝜉] = 𝑏+𝑐
2 .
A fuzzy variable 𝜉 is called normally distributed if it has a normal membership function
𝜋∣𝑥 − 𝑒∣ −1
𝜇(𝑥) = 2(1 + exp( √
)) , 𝑥 ∈ ℛ, 𝜎 > ′
6𝜎
(6)
By theorem 1,the expected value is 𝑒.
The definition of expected value operator is also applicable to discrete case. Assume that 𝜉 be a simple fuzzy variable
whose membership function is given by
⎧
𝜇1 , 𝑖𝑓 𝑥 = 𝑎1


⎨
𝜇2 , 𝑖𝑓 𝑥 = 𝑎2
𝜇(𝑥) =
...


⎩
𝜇𝑚 , 𝑖𝑓 𝑥 = 𝑎𝑚
where 𝑎1 , 𝑎2 , ..., 𝑎𝑚 are distinct numbers.Note that 𝜇1 ∨ 𝜇2 ∨ ... ∨ 𝜇𝑚 = 1. Then the expexted value of 𝜉 is
𝐸[𝜉] =
𝑚
∑
𝜔𝑖 𝑎𝑖 .
𝑖=1
where the weights are given by
𝜔𝑖 =
1
( max {𝜇𝑗 ∣𝑎𝑗 ≤ 𝑎𝑖 } − max {𝜇𝑗 ∣𝑎𝑗 < 𝑎𝑖 } + max {𝜇𝑗 ∣𝑎𝑗 ≥ 𝑎𝑖 } − max {𝜇𝑗 ∣𝑎𝑗 > 𝑎𝑖 })
1≤𝑗≤𝑚
1≤𝑗≤𝑚
1≤𝑗≤𝑚
2 1≤𝑗≤𝑚
for 𝑖 = 1, 2, ..., 𝑚. It is easy to verity that all 𝜔𝑖 ≥ 0 and the sum of all weights is just 1.
2.3 Some formulas variance of fuzzy variables
The variance of a fuzzy variable provides a measure of the spread of the distribution around its expected value. A
small value of variance indicates that the fuzzy variable is tightly concentrated around its expected value; and a large
value of variance indicates that the fuzzy variable has a wide spread around its expected value.
Definition 5 (Liu and Liu [4]) Let 𝜉 be a fuzzy variable with finite expected value 𝑒 .Then the variance of 𝜉 is
defined by
𝑉 [𝜉] = 𝐸[(𝜉 − 𝑒)2 ].
Definition 6 (Liu [12]) Let 𝜉 be a fuzzy variable, and k a positive number. Then the expected value 𝐸[𝜉 𝑘 ] is called
the 𝑘𝑡ℎ moment.
Theorem 2 Let 𝜉 be a normal fuzzy variable with membership function (6).Then the variance is 𝜎 2 .
Proof 𝐸[𝜉] = 𝑒,
√
√
𝐶𝑟{(𝜉 − 𝑒)2 ≥ 𝑟} = 𝐶𝑟({(𝜉 − 𝑒) ≥√ 𝑟} ∪ {(𝜉 − 𝑒) ≤ − 𝑟})
√
= 𝐶𝑟{(𝜉 − 𝑒) ≥ √𝑟} ∨ 𝐶𝑟{(𝜉 − 𝑒) ≤ − 𝑟}
= 𝐶𝑟{(𝜉 − 𝑒)√≥ 𝑟}
𝜋 𝑟 −1
= (1 + exp √
)
6𝜎
∫
then the variance
𝑉 [𝜉] =
∞
𝑒
√
𝜋 𝑟
(1 + exp √ )−1 𝑑𝑟 = 𝜎 2
6𝜎
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Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables
489
Theorem 3 Let 𝜉 be a exponential fuzzy variable with membership function (4).Then the second moment is 𝑚2 .
Theorem 4 Let 𝜉 be a triangular fuzzy variable (a,b, c) ,then its variance
⎧
33𝛼3 +11𝛼𝛽 2 +21𝛼2 𝛽−𝛽 3

, 𝛼>𝛽
⎨ 2
384𝛼
𝛼
𝑉 [𝜉] =
(7)
,
𝛼=𝛽
6

⎩ 33𝛽 3 +11𝛼2 𝛽+21𝛼𝛽 2 −𝛼3 , 𝛼 < 𝛽
384𝛽
where 𝛼 = 𝑏 − 𝑎, 𝛽 = 𝑐 − 𝑏.
Proof Let 𝑚 = 𝐸[𝜉],when 𝛼 = 𝛽,then 𝑚 = 𝑏 and
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟} =
∫
then
2
𝑉 [𝜉] = 𝐸[(𝜉 − 𝑚) ] =
{
0,
∞
0
√
𝛼− 𝑟
2𝛼 ,
0 ≤ 𝑟 ≤ 𝛼2
𝑟 ≥ 𝛼2
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 2}𝑑𝑟 =
when 𝛼 > 𝛽, 𝑚 < 𝑏
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟} = 𝐶𝑟{(𝜉 − 𝑚) ≥
√
𝛼2
6
√
𝑟} ∨ 𝐶𝑟{(𝜉 − 𝑚) ≤ − 𝑟}
(i) 0 ≤ 𝑟 ≤ (𝑏 − 𝑚)2 ,then
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟}
(ii)when (𝑏 − 𝑚)2 ≤ 𝑟 ≤ 𝑟𝑠 ,werer 𝑟𝑠 =
√
= 𝐶𝑟{𝜉 − 𝑚 √
≥ 𝑟}
= 12 [1 +
1 − 𝑟+𝑚−𝑎
]
2𝛼
√
𝑟+𝑚−𝑏+𝛼
= 1−
2𝛼
(𝛼+𝛽)2
16 ,then
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟} =
=
=
𝐶𝑟{𝜉 √
−𝑚≥
1 𝑚+ 𝑟−𝑏
[
]
2
−𝛽
√
𝑟}
√
−𝑚− 𝑟+𝑏+𝛽
2𝛽
(iii) when 𝑟𝑠 ≤ 𝑟 ≤ (𝑏 − 𝛼 − 𝑚)2 ,then
√
𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟} = 𝐶𝑟{𝜉 √
− 𝑚 ≤ − 𝑟}
= 12 [√𝑚− 𝛼𝑟−𝑎 ]
= − 𝑟+𝑚−𝑏+𝛼
2𝛼
(iv) when 𝑟 ≥ (𝑏 − 𝛼 − 𝑚)2 , 𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟} = 0 then,
∫∞
𝑉 [𝜉] = 𝐸[(𝜉 − 𝑚)2 ] = 0 𝐶𝑟{(𝜉 − 𝑚)2 ≥ 𝑟}𝑑𝑟
3
2
+21𝛼2 𝛽−𝛽 3
= 33𝛼 +11𝛼𝛽
384𝛼
when 𝛼 > 𝛽,a similar way may prove the equation (7) .The theorem 4 is proved.
Theorem 5 ([Chen 13]) Let 𝜉 be a trapezoidal fuzzy variable (a,b, c,d) ,Then its variance
(i) when 𝛼 = 𝛽,
𝑉 [𝜉] =
(ii)when 𝛼 > 𝛽,
{
𝑉 [𝜉] =
(iii) when 𝛼 < 𝛽,
𝑉 [𝜉] =
{
3(𝑏 − 𝑎 + 𝛽)2 + 𝛽 2
;
24
3
3
(𝑏−𝑚)3
1
− (𝑎−𝛼−𝑚)
+ (𝑏𝛼+𝑎𝛽−𝑚(𝛼+𝛽))
],
6 [−
𝛽
𝛼
𝛼𝛽(𝛼−𝛽)2
3
(𝑏−𝑚)3
(𝑎−𝛼−𝑚)3
(𝑏𝛼+𝑎𝛽−𝑚(𝛼+𝛽))3
1 (𝑎−𝑚)
],
− 𝛽 −
+
6[
𝛼
𝛼
𝛼𝛽(𝛼−𝛽)2
3
1 (𝑎−𝑚)
6[
𝛼
3
1 (𝑎−𝑚)
[
6
𝛼
+
−
3
(𝑏+𝛽−𝑚)3
− (𝑏𝛼+𝑎𝛽−𝑚(𝛼+𝛽))
],
𝛽
𝛼𝛽(𝛼−𝛽)2
(𝑏−𝑚)3
(𝑏+𝛽−𝑚)3
(𝑏𝛼+𝑎𝛽−𝑚(𝛼+𝛽))3
+
+
],
𝛽
𝛽
𝛼𝛽(𝛼−𝛽)2
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𝑎−𝑚<0
𝑎−𝑚≥0
𝑏−𝑚>0
𝑏−𝑚≤0
490
International Journal of NonlinearScience,Vol.9(2010),No.4,pp. 486-492
Theorem 6 Let 𝜉 be a triangular fuzzy variable (a,b, c) ,Then 𝑉 [𝜉] is an increasing function of 𝛼 and 𝛽.where
𝛼 = 𝑏 − 𝑎, 𝛽 = 𝑐 − 𝑏.
Proof (i) when 𝛼 = 𝛽,the equation holds trivially.
(ii) when 𝛼 > 𝛽, set 𝛼1 < 𝛼2 ,then
33𝛼23 + 11𝛼2 𝛽 2 + 21𝛼22 𝛽 − 𝛽 3
33𝛼13 + 11𝛼1 𝛽 2 + 21𝛼12 𝛽 − 𝛽 3
−
<0
384𝛼1
384𝛼2
𝑉 [𝜉] is an increasing function of 𝛼.
On the other hand,
22𝛼𝛽 + 21𝛼2 − 3𝛽 2
33𝛼3 + 11𝛼𝛽 2 + 21𝛼2 𝛽 − 𝛽 3 ′
) =
>0
(
384𝛼
384𝛼
𝑉 [𝜉] is an increasing function of 𝛽.
(iii) A similar way may prove the result. The theorem 6 is proved.
Theorem 7 Let 𝜉 be a trapezoidal fuzzy variable (a,b, c,d) .Then 𝑉 [𝜉] is an increasing function of 𝛼 and 𝛽.It is also
an increasing function of 𝑐 − 𝑏. where 𝛼 = 𝑏 − 𝑎, 𝛽 = 𝑑 − 𝑐
Proof (1) when 𝛼 = 𝛽,the equation holds trivially.
(2) where 𝛼 > 𝛽,Let ℎ = 𝑏 − 𝑎,then
𝑎−𝑚=
1
1
(𝛼 − 𝛽 − 2ℎ, 𝑏 − 𝑚 = (𝛼 − 𝛽 + 2ℎ,
4
4
1
1
𝑎 − 𝛼 − 𝑚 = − (3𝛼 + 𝛽 + 2ℎ), 𝑏𝛼 + 𝑎𝛽 − 𝑚(𝛼 + 𝛽) = (𝛼 + 𝛽 + 2ℎ)(𝛼 − 𝛽)
4
4
when 𝑎 − 𝑚 < 0
𝑉 [𝜉] =
1
[3(2ℎ + 𝛼)2 + 𝛽 2 + (3𝛼 + 𝛽 + 2ℎ)2 + (𝛼 + 𝛽 + 2ℎ)2 + (3𝛼 + 𝛽 + 2ℎ)(𝛼 + 𝛽 + 2ℎ)]
32
when 𝑎 − 𝑚 ≥ 0,
𝑉 [𝜉] =
1
(60𝛼ℎ2 + 66𝛼2 ℎ + 33𝛼3 + 11𝛼𝛽 2 + 21𝛼2 𝛽 + 36𝛼𝛽ℎ − 8ℎ3 − 6𝛽 2 ℎ − 12𝛽ℎ2 − 𝛽 3 )
64𝛼
It is easy to prove that it is an increasing function of 𝛼, 𝛽 and h,respectively.
(3) when 𝛼 < 𝛽,a slmilar way may prove the result.The theorem 7 is proved.
2.4 Expected value and variance of Geometric Liu process
Definition 7 (Liu [14])A fuzzy process Ct is said to be a Liu process if
(i) 𝐶0 = 0,
(ii) 𝐶𝑡 has stationary and independent increments,
(iii) every increment 𝐶𝑠+𝑡 − 𝐶𝑠 is a normally distributed fuzzy variable with expected value 𝑒𝑡 and variance 𝜎 2 𝑡2 .
The parameters 𝑒 and 𝜎 2 are called the drift and diffusion coefficients,respectively. Liu process is said to be standard
if 𝑒 = 0 and 𝜎 = 1.
Definition 8(Liu [14])Let 𝐶𝑡 be a standard Liu process. Then the fuzzy process 𝐺𝑡 = exp(𝑒𝑡 + 𝜎𝐶𝑡 ) is called a
geometric Liu process.
In 2008, Li [15] proved that 𝐺𝑡 has a lognormal membership function and obtained its expeceted value and variance.In
addition, alternative fuzzy stock models were introduced by Peng [16].
Theorem 8 Let 𝐺𝑡 be a geometric Liu process whose membership function
𝜋∣ ln 𝑥 − 𝑒𝑡∣ −1
√
)) , 𝑥 > 0
6𝜎𝑡
√
√
√
then it has expected value 𝐸[𝐺𝑡 ] = exp(𝑒𝑡) csc( 6𝜎𝑡) 6𝜎𝑡, ∀𝑡 < 𝜋( 6𝜎)−1 .
𝜇(𝑥) = 2(1 + exp (
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(8)
Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables
491
√
Proof ∀𝑡 < 𝜋( 6𝜎)−1
∫∞
𝐸[𝐺𝑡 ] = 0 𝐶𝑟{𝐺𝑡 ≥ 𝑟}𝑑𝑟
∫∞
∫ exp(𝑒𝑡)
𝑥−𝑒𝑡) −1
𝑥) −1
√
√
= 0
[1 − (1 + exp( 𝜋(𝑒𝑡−ln
))
]𝑑𝑥
+
(1 + exp( 𝜋(ln
)) 𝑑𝑥
exp(𝑒𝑡)
6𝜎𝑡
6𝜎𝑡
𝜋𝑒
𝜋𝑒
∫ exp(𝑒𝑡)
∫∞
exp( √6𝜎 )
exp( √6𝜎 )
= 0
√ 𝜋 ) 𝑑𝑥 + exp(𝑒𝑡)
√ 𝜋 ) 𝑑𝑥
𝜋𝑒
𝜋𝑒
=
∫∞
0
exp( √6𝜎 )+𝑥
exp( √𝜋𝑒
)
6𝜎
√𝜋 )
6𝜎𝑡
exp( √𝜋𝑒
)+𝑥
6𝜎
∫∞
exp(𝑒𝑡) 0 (1
√+
6𝜎𝑡
exp( √6𝜎 )+𝑥
6𝜎𝑡
𝑑𝑥
√𝜋
6𝜎𝑡
𝑥 √ )−1 𝑑𝑥
=
= exp(𝑒𝑡) csc( 6𝜎𝑡) 6𝜎𝑡
Example 5 Let 𝐺𝑡 be a geometric Liu process.Li X [15] obtained the variance formula
If 𝑡 ≥ (2√𝜋6𝜎) ,
𝑉 [𝐺𝑡 ] =
∫ 𝐸 2 [𝐺𝑡 ]−𝑒𝑥𝑝(2𝑒𝑡)
√
𝜋
(1 + exp (1 + exp ( √6𝜎𝑡
(𝑒𝑡 − ln(𝐸[𝐺𝑡 ] − 𝑥))))−1 𝑑𝑥
0∫
√
∞
𝜋
+ 𝐸 2 [𝐺𝑡 ]−𝑒𝑥𝑝(2𝑒𝑡) (1 + exp ( √6𝜎𝑡
(ln(𝐸[𝐺𝑡 ] + 𝑥) − 𝑒𝑡)))−1 𝑑𝑥
Otherwise,𝑉 [𝐺𝑡 ] = +∞.
3
Conclusions
The main contribution of the present paper is to obtain the expected value and variance formulas of the triangular fuzzy
variable,the trapezoidal fuzzy variable , normal fuzzy variable and geometric Liu process. On the other hand, some
properities of variance are griven.
Acknowledgments
This work is supported by the institution of higher learning scientific research Program (No.Hjkj200908) of Hainan
Provincial Department of Education, the Hainan Natural Science Foundation(No.807025),China.
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