Formulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable

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1 1 Formulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable Xiumei Chen 1,, Yufu Ning 1,, Xiao Wang 1, 1 School of Information Engineering, Shandong Youth University of Political Science, Jinan 513, China Key Laboratory of Information Security and Intelligent Control in Universities of Shandong, Jinan 513, China cxm@sdyu.edu.cn Abstract Uncertainty theory is a branch of mathematics that deals with human uncertainty, and uncertain variable is used to model an uncertain quantity. Uncertainty exists not only in real quantities but also in complex quantities. Complex uncertain variable is mainly used to model a complex uncertain quantity. In uncertainty theory, inverse uncertainty distribution provides an easy way to calculate expected value as well as variance of an uncertain variable. This paper proposes formulas to calculate variance and pseudovariance via the inverse uncertainty distributions of the real and imaginary parts of a complex uncertain variable. Besides, an inequality about variance and pseudo-variance of a complex uncertain variable is also derived. Keywords: uncertain variable, complex uncertain variable, inverse uncertainty distribution, variance, pseudo-variance 1 Introduction Probability theory, as a branch of axiomatic mathematics for modeling indeterminacy, was founded by Kolmogorov [3] in Indices in probability theory such as expected value and variance are often used as criteria to analyse and deal with indeterminacy problems. Until now probability theory has been developed steadily and applied widely in science and engineering. As is known to us all, a fundamental premise of applying probability theory is that the estimated probability distribution is close enough to the long-run cumulative frequency. When the sample size is large enough, it is possible to believe that the estimated probability distribution is close enough to the long-run cumulative frequency. However, in many cases, no samples are available to estimate a probability distribution. In this case, we have no choice but to invite some domain experts to evaluate the belief degrees that possible events happen. Kahneman and Tversky [] showed that human beings usually overweight unlikely events. From another side, Liu [14] showed that human beings usually estimate a much wider range of values than the object actually takes. As a result, the probability theory is not applicable in this case. Otherwise if modeling belief degrees by probability theory, counterintuitive results may be led and a counterexample may be found in Liu [11]. In order to deal with belief degree mathematically, Liu proposed uncertainty theory in 7 [4] by uncertain measure and refined it in 1 [9]. It is a branch of axiomatic mathematics basing on normality, duality, subadditivity, and product axioms. So far, the uncertain theory has been developed many branches including uncertain programming [6], [15]), uncertain risk analysis [8]), uncertain logic [1]), uncertain process [5]), uncertain finance [1], [1]), etc. In uncertain theory, the concept of uncertain variable was proposed to present an uncertain quantity and the concept of uncertainty distribution was first introduced to describe uncertain variable. After that, Peng and Iwamura [18] verified a sufficient and necessary condition for a function being the uncertainty

2 distribution. In order to better describe an uncertain variable, Liu [9] introduced the concept of the inverse uncertainty distribution of an uncertain variable. For independent uncertain variables with regular uncertainty distributions, Liu [9] provided some operational laws for calculating inverse uncertainty distributions. In order to rank the uncertain variables, the concept of expected value operator was proposed by Liu [4]. In addition, Liu and Xu [16] gave some inequalities on expected value operator. And then Liu [9] gave a formula to calculate the expected value via its inverse uncertainty distribution and the formula was generalized by Liu and Ha [17] in 1. Variance of an uncertain variable is another concept proposed by Liu [4] which provides a degree of the spread of the distribution around its expected value. In order to calculate the variance of an uncertain variable, Liu [9] gave a stipulation and by which Yao [] gave a calculation formula via inverse uncertainty distribution. As an extension of uncertain variable, the concept of uncertain vector was defined by Liu [4]. In addition, Liu [13] discussed the independence of uncertain vectors. Real quantity is modeled by uncertain variable in uncertain theory. However, uncertainty not only appear in the real quantities but also in complex quantities. In order to model complex uncertain quantities, Peng [19] presented the concepts of complex uncertain variable and complex uncertainty distribution, and proved the sufficient and necessary condition for complex uncertainty distribution. Besides, the expected value and variance were proposed to model complex uncertain variables. In this paper, we will further study the variance of a complex uncertain variable and propose the concept of pseudo-variance. It mainly provides a formula to calculate the variance as well as the pseudo-variance via inverse uncertainty distributions of the real and imaginary parts of complex uncertain variable. The rest of this paper is organized as follows. In Section, we review some basic concepts and theorems about uncertain variables. And Section 3 introduces some concepts and theorems about complex uncertain variables. Then some theorems about variance will be showed in Section 4. Pseudo-variance will be introduced and a stipulation to calculate pseudo-variance will be displayed in Section 5. After that, an inequality about variance and pseudo-variance will be proved in Section 6. Finally, some remarks will be made in Section 7. Preliminary In this section, some basic concepts and theorems in uncertainty theory are introduced, which are used throughout this paper. Definition.1 Liu [4]) Let L be a σ-algebra on a nonempty set Γ. A set function M is called an uncertain measure if it satisfies the following axioms: Axiom 1. Normality Axiom) M{Γ} 1; Axiom. Duality Axiom) M{Λ} + M{Λ c } 1 for any Λ L; Axiom 3. Subadditivity Axiom) For every countable sequence of {Λ i } L, we have { } M Λ i M{Λ i }. i1 The triplet Γ, L, M) is called an uncertainty space, and each element Λ in L is called an event. In order to obtain an uncertain measure of compound event, a product uncertain measure is defined by Liu [7] as follows: Axiom 4. Product Axiom) Let Γ k, L k, M k ) be uncertainty spaces for k 1,,. The product uncertain measure M is an uncertain measure satisfying { } M Λ k M k {Λ k } where Λ k are arbitrarily chosen events from L k for k 1,,, respectively. k1 i1 k1

3 3 Definition. Liu [4]) An uncertain variable ξ is a measurable function from an uncertainty space Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. {ξ B} {γ Γ ξγ) B} Definition.3 Liu [4]) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φx) M {ξ x}, x R. Definition.4 Liu [9]) An uncertainty distribution Φx) is said to be regular if it is a continuous and strictly increasing function with respect to x at which < Φx) < 1, and lim Φx), lim x Φx) 1. x + In addition, the inverse function Φ 1 α) is called the inverse uncertainty distribution of ξ. An uncertain variable ξ is said to be linear if it has a linear uncertainty distribution, if x < a Φx) x a)/b a), if a x b 1, if x > b which is denoted by ξ La, b). Apparently, the linear uncertain variable ξ is regular, and has an inverse uncertainty distribution Φ 1 α) αb a) + a. An uncertain variable ξ is said to be normal if it has a normal uncertainty distribution Φx) )) 1 πe x) 1 + exp, x R 3σ denoted by ξ Ne, σ) where e and σ are real numbers with σ >. The normal uncertain variable is regular, and the inverse uncertainty distribution is Φ 1 α) e + σ 3 1 α. Definition.5 Liu [7]) The uncertain variables ξ 1, ξ,, ξ n are said to be independent if { n } n M ξ i B i ) M {ξ i B i } i1 for any Borel sets B 1, B,, B n of real numbers. Theorem.1 Liu [9]) Assume ξ 1, ξ,, ξ n are independent uncertain variables with regular uncertainty distributions Φ 1, Φ,, Φ n, respectively. If the function fx 1, x,, x n ) is strictly increasing with respect to x 1, x,, x m and strictly decreasing with respect to x m+1, x m+,, x n, then ξ fξ 1, ξ,, ξ n ) has an inverse uncertainty distribution Ψ 1 α) f Φ 1 1 α),, Φ 1 m α), Φ 1 m+1 1 α),, Φ 1 n 1 α) ). i1

4 4 Definition.6 Liu [4]) Let ξ be an uncertain variable. The expected value of ξ is defined by + E[ξ] M{ξ r}dr M{ξ r}dr provided that at least one of the above two integrals is finite. In 1, Liu [9] first gave the formula to calculate the expected value via inverse uncertainty distribution. The formula is E[ξ] Φ 1 α)dα, where Φ 1 is the inverse uncertainty distribution of ξ. By using this formula, the conclusion that the expected value of a linear uncertain variable ξ La, b) is E[ξ] a+b, and the expected value of a normal uncertain variable η Ne, σ) is E[η] e was obtained. Theorem. Liu and Ha [17]) Assume ξ 1, ξ,, ξ n are independent uncertain variables with regular uncertainty distributions Φ 1, Φ,, Φ n, respectively. If the function fx 1, x,, x n ) is strictly increasing with respect to x 1, x,, x m and strictly decreasing with respect to x m+1, x m+,, x n, then ξ fξ 1, ξ,, ξ n ) has an expected value E[ξ] f Φ 1 1 α),, Φ 1 m α), Φ 1 m+1 1 α),, Φ 1 n 1 α) ) dα. Based on the above result, Liu [9] proved the linearity property of the expected value operator. Theorem.3 Liu [9]) Assume ξ and η are two independent uncertain variables. Then for any real numbers a and b, we have E[aξ + bη] ae[ξ] + be[η]. In 7, Liu [4] gave the definition of the variance of uncertain variable as follows. Definition.7 Liu [4]) Let ξ be an uncertain variable with a finite expected value E[ξ]. Then the variance of ξ is defined by V [ξ] E[ξ E[ξ]) ]. In 14, Yao [] gave a formula to calculate the variance via inverse uncertainty distribution, that is V [ξ] Φ 1 α) E[ξ]) dα. By using this formula, the variance of a linear uncertain variable ξ La, b) is V [ξ] b a) 1, and the variance of a normal uncertain variable η Ne, σ) is V [η] σ. Theorem.4 Liu [9]) Let ξ be an uncertain variable with a finite expected value E[ξ]. Then for any real numbers a and b, we have V [aξ + b] a V [ξ]. 3 Complex uncertain variable In this section, we introduce some concepts and theorems of complex uncertain variables which were first proposed by Peng [19] in 1. As a complex function on uncertainty space, complex uncertain variable is mainly used to model complex uncertain quantities.

5 5 Definition 3.1 Peng [19]) A complex uncertain variable is a measurable function ζ from an uncertainty space Γ, L, M) to the set of complex numbers, i.e., for any Borel set B of complex numbers, the set is an event. {ζ B} {γ Γ ζγ) B} Theorem 3.1 Peng [19]) A variable ζ from an uncertainty space Γ, L, M) to the set of complex numbers is a complex uncertain variable if and only if Reζ and Imζ are uncertain variables where Reζ and Imζ represent the real and the imaginary parts of ζ, respectively. Definition 3. Peng [19]) The complex uncertainty distribution Φx) of a complex uncertain variable ζ is a function from C to [, 1] defined by for any complex number c. Φc) M{Reζ) Rec), Imζ) Imc)} Theorem 3. Peng [19]) A function Φc) : C [, 1] is a complex uncertainty distribution if and only if it is increasing with respect to the real part Rec) and the imaginary part Imc) such that i) lim Φx + bi) 1, lim x ii) lim x +,y + Φx + yi). Φa + yi) 1, for any a, b R; y Definition 3.3 Peng [19]) The complex uncertain variables ζ 1, ζ,, ζ n are said to be independent if { n } n M ζ i B i ) M {ζ i B i } i1 for any Borel sets B 1, B,, B n of complex numbers. In order to model complex uncertain variable, the expected value is proposed as below. Definition 3.4 Peng [19]) Let ζ be a complex uncertain variable. The expected value of ζ is defined by i1 E[ζ] E[Reζ)] + ie[imζ)] provided that E[Reζ)] and E[Imζ)] are finite, where E[Reζ)] and E[Imζ)] are expected values of uncertain variables Reζ) and Imζ), respectively. A complex uncertain variable ζ is said to be linear if Reζ) and Imζ) are both linear uncertain variables. Consider the linear complex uncertain variable ζ La 1, b 1 ) + ila, b ). We have E[ζ] a 1 + b 1 + i a + b. A complex uncertain variable ζ is said to be normal if Reζ) and Imζ) are both normal uncertain variables. Consider the normal complex uncertain variable ζ Ne 1, σ 1 ) + ine, σ ). We have E[ζ] e 1 + ie. Theorem 3.3 Peng [19]) Assume that ζ and τ are independent complex uncertain variables such that E[ζ] and E[τ] exist. Then for any complex numbers α and β, E[αζ + βτ] exists and E[αζ + βτ] αe[ζ] + βe[τ]. Definition 3.5 Peng [19]) Let ζ be a complex uncertain variable with expected value E[ζ]. variance of ζ is defined by V [ζ] E[ ζ E[ζ] ]. Then the

6 6 4 Some theorems In this section, we first give a formula to calculate the expected value of a complex uncertain variable. Subsequently, a stipulation to calculate variance of a complex uncertain variable is presented. Theorem 4.1 Let ζ be a complex uncertain variable. Assume Reζ) and Imζ) are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then E[ζ] [Φ 1 α) + iψ 1 α)]dα. Proof: By Definition 3.4 and Theorem.1, we immediately get E[ζ] E[Reζ)] + ie[imζ)] [Φ 1 α) + iψ 1 α)]dα. The linearity expression of expected value via inverse uncertainty distribution is as follows. Theorem 4. Let ζ and τ be independent complex uncertain variables such that E[ζ] and E[τ] exist. Assume Reζ), Imζ), Reτ) and Imτ) are independent uncertain variables with regular uncertainty distributions Φ 1, Ψ 1, Φ and Ψ, respectively. Then for any complex numbers α and β, E[αζ + βτ] exists and E[αζ + βτ] α [Φ 1 1 α) + iψ 1 1 α)] + β Proof: The theorem follows from Theorem 3.3 and Theorem 4.1. [Φ 1 α) + iψ 1 α)]dα. Definition 4.1 Let ζ be a complex uncertain variable. The complex conjugate uncertain variable of ζ ξ +iη is defined as ζ ξ iη. The modulus of ζ can be represented by ζ ζζ ξ + η. By the definition of complex conjugate uncertain variable, we can rewrite the variance of ζ as V [ζ] E[ζ E[ζ])ζ E[ζ]) ]. Theorem 4.3 If ζ is a complex uncertain variable with finite expected value, α and β are complex numbers, then V [αζ + β] α V [ζ]. Proof: It follows from the definition of variance that V [αζ + β] E[ αζ + β E[αζ + β] ] α E[ ζ E[ζ] ] α V [ζ]. Since the uncertain measure is a subadditivity measure, the variance of complex uncertain variable ζ cannot be derived by the uncertainty distribution. A stipulation of variance of ζ with inverse uncertainty distribution of the real and imaginary parts of ζ is presented as follows. Stipulation 1 Let ζ ξ + iη be a complex uncertain variable with the real part ξ and imaginary part η. The expected value of ζ exists and E[ζ] E[ξ] + ie[η]. Assume ξ and η are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then V [ζ] [Φ 1 α) E[ξ]) + Ψ 1 α) E[η]) ]dα.

7 7 Example 1 Consider the linear complex uncertain variable ζ ξ + iη and ξ La 1, b 1 ), η La, b ). Then the inverse uncertainty distributions of ξ and η are Φ 1 α) b 1 a 1 )α + a 1 and Ψ 1 α) b a )α + a, respectively. And the expected values are E[ξ] a1+b1 and E[η] a+b, respectively, so that V [ζ] [Φ 1 α) E[ξ]) + Ψ 1 α) E[η]) ]dα [b 1 a 1 )α + a 1 a 1 + b 1 [b 1 a 1 ) α 1 ) + b a ) α 1 ) ]dα ) + b a )α + a a + b ) ]dα b 1 a 1 ) + b a ). 1 Example Consider the normal complex uncertain variable ζ ξ+iη and ξ Ne 1, σ 1 ), η Ne, σ ). Then the inverse uncertainty distributions of ξ and η are Φ 1 α) e 1 + σ1 3 π ln α 1 α and Ψ 1 α) e + σ 3 π ln α 1 α, respectively. And the expected values are E[ξ] e 1 and E[η] e, respectively, so that V [ζ] [e 1 + σ α e 1) + e + σ 3 1 α e ) ]dα [ σ α ) + σ 3 1 α ) ]dα 3σ 1 + σ ) π σ 1 + σ. ln α 1 α ) dα Example 3 Consider the complex uncertain variable ζ ξ +iη and ξ La, b), η Ne, σ). Then the inverse uncertainty distributions of ξ and η are Φ 1 α) b a)α + a and Ψ 1 α) e + σ 3 π ln α 1 α, respectively. And the expected values are E[ξ] a+b and E[η] e, respectively, so that V [ζ] [b a)α + a a + b )) + e + σ 3 1 α e) ]dα [b a)α + a b ) + σ 3 1 α ) ]dα [b a) α 1 ) + 3σ π ln α 1 α ) ]dα b a) + σ. 1 Theorem 4.4 Let ζ ξ + iη be a complex uncertain variable with the real part ξ and imaginary part η. The expected value of ζ exists and E[ζ] E[ξ] + ie[η]. Assume ξ and η are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then Proof: V [ζ] [Φ 1 α)) + Ψ 1 α)) ]dα E[ξ] E[η]. For simplicity, we use a + ib to denote the expected value E[ζ]. It follows from Stipulation 1 that V [ζ] [Φ 1 α) a) + Ψ 1 α) b) ]dα [Φ 1 α)) aφ 1 α) + a + Ψ 1 α)) bψ 1 α) + b ]dα [Φ 1 α)) + Ψ 1 α)) ]dα a Φ 1 α)dα b Ψ 1 α)dα + a + b.

8 8 Noting that the expected value of ξ is just E[ξ] Φ 1 α)dα a and the expected value of η is just E[η] Ψ 1 α)dα b, we have V [ζ] The theorem is thus proved. [Φ 1 α)) + Ψ 1 α)) ]dα a b + a + b [Φ 1 α)) + Ψ 1 α)) ]dα a b. Theorem 4.5 Let ζ and τ be two independent complex uncertain variables. Assume ζ ξ 1 + iη 1 and τ ξ + iη. Suppose that uncertain variables ξ 1, η 1, ξ and η are independent with regular uncertainty distributions Φ 1, Ψ 1, Φ, and Ψ, respectively. Then we have V [ζ + τ] V [ζ] + V [τ]), and the equality holds if and only if there exist two real numbers µ and ν such that Φ 1 x) Φ x + µ) and Ψ 1 x) Ψ x + ν). Proof: Since ζ + τ ξ 1 + iη 1 + ξ + iη ξ 1 + ξ + iη 1 + η ), it follows from Theorem.1 that ξ 1 + ξ and η 1 + η have inverse uncertainty distributions Φ Φ 1 and Ψ Ψ 1, respectively. In addition, by the linearity of expected operator for independent uncertain variables, we have By Stipulation 1, we obtain V [ζ + τ] Φ 1 1 α) + Φ 1 E[ζ + τ] E[ξ 1 ] + E[ξ ]) + ie[η 1 ] + E[η ]). α) E[ξ 1] + E[ξ ])) + Ψ 1 1 α) + Ψ 1 α) E[η 1] + E[η ])) ) dα [Φ 1 1 α) E[ξ 1]) + Φ 1 α) E[ξ ])] + [Ψ 1 1 α) E[η 1]) + Ψ 1 α) E[η ])] ) dα Φ 1 1 α) E[ξ 1]) + Φ 1 α) E[ξ ]) + Ψ 1 1 α) E[η 1]) + Ψ 1 [Φ 1 1 α) E[ξ 1]) + Φ 1 α) E[ξ ]) ]dα + V [ζ] + V [τ]). Besides, note that the equality holds if and only if Φ 1 1 α) E[ξ 1] Φ 1 α) E[ξ ] and Ψ 1 1 α) E[η 1] Ψ 1 α) E[η ]. α) E[η ]) ) dα ) [Ψ 1 1 α) E[η 1]) + Ψ 1 α) E[η ]) ]dα Write µ E[ξ ] E[ξ 1 ] and ν E[η ] E[η 1 ]. Then we have Φ 1 α) Φ 1 1 α) µ and Ψ 1 α) Ψ 1 1 α) ν. So we have Φ 1 x) Φ x + µ) and Ψ 1 x) Ψ x + ν). Thus the theorem is verified. 5 Pseudo-variance The definition of Pseudo-variance is initiated and a stipulation to calculate pseudo-variance of a complex uncertain variable is showed in this section. Definition 5.1 Let ζ be a complex uncertain variable with expected value E[ζ]. Then the pseudo-variance is defined by Ṽ [ζ] E[ζ E[ζ]) ].

9 9 Stipulation Let ζ ξ + iη be a complex uncertain variable with the real part ξ and imaginary part η. The expected value of ζ exists and E[ζ] E[ξ] + ie[η]. Assume ξ and η are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then Ṽ [ζ] [Φ 1 α) E[ξ]) Ψ 1 α) E[η]) ] + i[φ 1 α) E[ξ])Ψ 1 α) E[η])] ) dα. Example 4 Consider the linear complex uncertain variable ζ ξ + iη, and ξ La 1, b 1 ), η La, b ). By Stipulation, we have Ṽ [ζ] [Φ 1 α) E[ξ]) Ψ 1 α) E[η]) ] + i[φ 1 α) E[ξ])Ψ 1 α) E[η])] ) dα b 1 a 1 )α + a 1 a 1 + b 1 ) b a )α + a a + b ) +i[b 1 a 1 )α + a 1 a 1 + b 1 )b a )α + a a + b )]dα α 1 ) [b 1 a 1 ) b a ) + ib 1 a 1 )b a )]dα [b 1 a 1 ) + ib a )] α 1 ) dα [b 1 a 1 ) + ib a )]. 1 Example 5 Consider the normal complex uncertain variable ζ ξ + iη, and ξ Ne 1, σ 1 ), η Ne, σ ). We have Ṽ [ζ] [e 1 + σ α e 1) e + σ 3 1 α e ) ] +i[e 1 + σ α e 1)e + σ 3 1 α e )]dα [ σ α ) σ 3 1 α ) ] + i σ α )σ 3 1 α )dα 3σ 1 σ ) 6iσ 1 σ π σ 1 σ + iσ 1 σ σ 1 + iσ ). ln α 1 α ) dα Example 6 Consider the complex uncertain variable ζ ξ + iη, and ξ La, b), η Ne, σ). We can get Ṽ [ζ] [b a)α + a a + b ) e + σ 3 1 α e) +ib a)α + a a + b )e + σ 3 1 α e)]dα [b a)α + a b ) σ 3 1 α ) ]dα + i [b a)α + a b 3 )σ 1 α )]dα [b a) α 1 ) 3σ π ln α 3σ 1 α ) ]dα + i [b a) π α 1 )ln α 1 α ]dα b a) α 1 ) dα 3σ π ln α 3σ 1 1 α ) dα + ib a) α 1 π )ln α 1 α dα b a) 3σ σ + ib a) 1 π.

10 1 Similar to Theorem 4.4, we derive another formula to calculate the pseudo-variance of a complex uncertain variable as below. Theorem 5.1 Let ζ ξ + iη be a complex uncertain variable with the real part ξ and imaginary part η. The expected value of ζ exists and E[ζ] E[ξ] + ie[η]. Assume ξ and η are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Thus Ṽ [ζ] [Φ 1 α) Ψ 1 α) ]dα E[ξ] + E[η] + i Φ 1 α)ψ 1 α)dα E[ξ]E[η]). Proof: For simplicity, we use a + ib to denote the expected value E[ζ]. It follows from Stipulation that Ṽ [ζ] +i +i[ [Φ 1 α) a) Ψ 1 α) b) ]dα + i [Φ 1 α) aφ 1 α) + a Ψ 1 α) + bψ 1 α) b ]dα [Φ 1 α)ψ 1 α) aψ 1 α) bφ 1 α) + ab]dα [Φ 1 α) Ψ 1 α) ]dα a Φ 1 α)ψ 1 α)dα a Φ 1 α)dα + b Ψ 1 α)dα b [Φ 1 α) a)ψ 1 α) b)]dα Ψ 1 α)dα + a b Φ 1 α)dα + ab]. Noting that the expected value of ξ is just E[ξ] Φ 1 α)dα a and the expected value of η is just E[η] Ψ 1 α)dα b, we have Ṽ [ζ] [Φ 1 α) Ψ 1 α) ]dα a + b + a b + i[ [Φ 1 α) Ψ 1 α) ]dα a + b + i Φ 1 α)ψ 1 α)dα ab + ab] Φ 1 α)ψ 1 α)dα ab). The theorem is thus proved. 6 An inequality about variance and pseudo-variance In this section, we will prove an inequality about variance and pseudo-variance of a complex uncertain variable. Theorem 6.1 Let ζ ξ + iη be a complex uncertain variable with the real part ξ and the imaginary part η. The expected value of ζ exists and E[ζ] E[ξ] + ie[η]. Assume ξ and η are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then we have Ṽ [ζ] V [ζ], and the equality holds if and only if there exist two real constants c and λ such that Ψx) Φλx + c). Proof: It follows from Stipulation that Ṽ [ζ] [Φ 1 α) E[ξ]) Ψ 1 α) E[η]) ] + i[φ 1 α) E[ξ])Ψ 1 α) E[η])] ) dα.

11 11 By Definition 4.1, we just need to prove [ Φ 1 α) E[ξ]) dα Ψ 1 α) E[η]) dα] + 4[ After squaring both sides and simplifying the equality, we have [ Φ 1 α) E[ξ])Ψ 1 α) E[η])dα] Φ 1 α) E[ξ])Ψ 1 α) E[η])dα] Φ 1 α) E[ξ]) dα + Φ 1 α) E[ξ]) dα Ψ 1 α) E[η]) dα. Ψ 1 α) E[η]) dα. 1) Inequality 1) holds by Cauchy-Schwarz Inequality if and only if there exists a real number λ such that Φ 1 α) E[ξ] λψ 1 α) E[η]). Write c E[ξ] λe[η], then we have Φ 1 α) λψ 1 α) + c, or equivalently, Ψx) Φλx + c). Remark Consider the linear complex uncertain variable and the normal complex uncertain variable. From Examples 1, 4, and 5, we can easily obtain Ṽ [ζ] V [ζ]. However, consider the complex uncertain variable ζ ξ + iη, and ξ La, b), η Ne, σ). By Examples 3 and 6, since a) 3σ Ṽ [ζ] b σ + ib a) 1 π < b a) 1 σ ) + b a) 3σ π ) b a) ) 1 + σ 4 + b a) σ 3 π 1 6 ) b a) 1 b a) 1 ) + σ 4 + b a) σ ) 6 + σ V [ζ], thus we have Ṽ [ζ] < V [ζ]. 7 Conclusions In this paper, we introduced the concept of pseudo-variance of a complex uncertain variable. Two formulas to calculate the variance and pseudo-variance were proposed via inverse uncertain distributions of the real and imaginary parts of complex uncertain variables. In addition, this paper gave an inequality about variance and pseudo-variance of a complex uncertain variable. Acknowledgements This work is supported by Natural Science Foundation of Shandong Province ZR14GL). References [1] Chen XW, Ameriacn option pricing formula for uncertain financial market, International Journal of Operations Research, Vol.8, No., pp.3-37, 11). [] Kahneman D, and Tversky A, Prospect theory: an analysis of decision under risk, Econometrica, Vol.47, No., pp. 63-9, 1979).

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