Variance and Pseudo-Variance of Complex Uncertain Random Variables

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1 Variance and Pseudo-Variance of Complex Uncertain andom Variables ong Gao 1, Hamed Ahmadzade, Habib Naderi 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran Abstract Variance of a complex uncertain random variable provides a degree of spread of the distribution around its expected value. This paper provides some practical quantities to describe a complex uncertain random variable. The typical one is the expected value which is the uncertain version of the center of gravity of a physical body. Moreover, there exist additional measures to describe a distribution in brief terms, such as the variance which measures the dispersion or spread, and pseudo-variance. For calculating the variance of a complex uncertain random variable, some formulas are derived via inverse uncertainty distribution. Furthermore, the relation between variance and pseudo-variance of a complex uncertain random variable is studied. Then the main results are explained by using several examples. Keywords: Chance theory, complex uncertain random variable, variance, pseudo-variance. 1 Introduction Probability theory is used to model objective randomness related to frequency. And the frequency is extracted from samples. Sometimes we come into contact with some samples including complex quantities, such as periodic signal, alternating current in electricity and two-dimensional potential flow in fluid mechanics. For modeling such phenomena, the concept of complex normal random variable was proposed by Wooding [15]. After that, the characteristic function of such random variable was studied by Turin [14]. Furthermore, Goodman [6] established the statistical properties of complex variables. Another different type of indeterminacy is uncertainty. In practice, we do not always have samples data or samples data are not reliable, such as financial affairs and new market in portfolio selection. In these situations, we have to reply on experts belief degrees. In order to deal with non-random phenomena, uncertainty theory was founded by Liu [9] as a branch of axiomatic mathematics based on normality, duality, subadditivity, and product axioms. In order to model complex quantities under uncertainty assumptions, Peng [13] introduced the concept of complex uncertain variables and investigated several properties of this concept such as complex uncertainty distribution, expected value, and variance. By invoking inverse uncertainty distribution, Chen et al. [] derived several formulas for calculating variance Corresponding author. 1

2 and pseudo-variance of complex uncertain variables. In addition, convergence of complex uncertain sequences was discussed by Ning and Wang [3]. However, randomness and uncertainty occur simultaneously in a complex system in many cases. In order to describe such a complex system, Liu [11] first established chance theory which is a mathematical methodology for modeling complex systems with both uncertainty and randomness including chance measure, uncertain random variable, chance distribution, operational law, expected value, and variance. After that, Liu [1] presented the operational law of uncertain random variable, and provided the formula for calculating expected value. As an application, Liu [1] proposed uncertain random programming as a branch of mathematical programming involving uncertain random variables. As we all know, the variance of uncertain random variable will provide a degree of the spread of the distribution around its expected value. Consequently, Several authors devoted their studies to variance of uncertain random variables. For instance, Guo and Wang [7] derived a formula for calculating the variance of uncertain random variables via uncertainty distribution. Gao and Wang [4] established a method for calculating moments via chance distribution. By invoking inverse uncertainty distribution, Ahmadzade et al. [1] provided several formulas for calculating variance and moments of uncertain random variables. In addition, in order to deal with complex quantities based on chance theory, Gao et al. [5] introduced complex uncertain random variables and studied some properties such as expected value, and variance. In this paper, by using inverse uncertainty distribution, we provide several formulas for calculating variance and pseudo-variance of complex uncertain random variables. The rest of this paper is organized as follows. In Section, some basic concepts of uncertainty theory and chance theory are reviewed. In Section 3, by invoking inverse uncertainty distribution, several formulas are derived for calculating variance and pseudo-variance of complex uncertain random variables. Then we propose and prove two inequalities about variance and pseudo-variance of complex uncertain random variables. Finally, we give a brief conclusion in Section 4. Preliminaries In this section, we review some concepts in uncertainty theory and chance theory, including uncertain variable, complex uncertain variable, chance distribution, operational law, expected value, and variance..1 Uncertain Variables and Complex Uncertain Variables In this subsection, we provide several definitions and elementary concepts of uncertainty theory that will be used in the next sections. For more details, the reader could refer to eferences [9, 1]. Definition 1 (Liu [9]) Let L be a σ-algebra on a nonempty set Γ. A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms: (i) (Normality Axiom) M{Γ} 1 for the universal set Γ. (ii) (Duality Axiom) M{Λ} + M{Λ c } 1 for any event Λ. (iii) (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ,, we have { } M Λ i M {Λ i }. i1 i1

3 Then the triple (Γ, L, M) is called uncertainty space. Next, the product uncertain measure was proposed by Liu [1] via the following axiom. (iv) (Product Axiom)(Liu [1]) 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 }, k1 k1 where Λ k are arbitrarily chosen events from L k for k 1,,, respectively. Definition (Liu [9]) An uncertain variable ξ is a function from an uncertainty space (Γ, L, M) to the set of real numbers such that {ξ B} is an event for any Borel set B of real numbers. Definition 3 (Liu [1]) 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 i1 for any Borel sets B 1, B,, B n of real numbers. Theorem 1 ([1]) Let ξ 1, ξ,, ξ n be independent uncertain variables, and f 1, f,, f n be measurable functions. Then f 1 (ξ 1 ), f (ξ ),, f n (ξ n ) are independent uncertain variables. Definition 4 (Liu [1]) Let ξ be an uncertain variable. Then its uncertainty distribution is defined as Φ(x) M{ξ x} for any x. Definition 5 (Liu [1]) An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which < Φ(x) < 1, and lim Φ(x), lim x Φ(x) 1. x Definition 6 (Liu [1]) Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). Then the inverse function Φ 1 (α) is called the inverse uncertainty distribution of ξ. Theorem (Liu [1]) Let ξ 1,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ,, Φ n, respectively. If f(x 1,, x n ) is strictly increasing with respect to x 1,, x m and decreasing with respect to x m+1,, x n then ξ f(ξ 1, ξ,, ξ n ) is an uncertain variable with 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). Definition 7 (Peng[13]) A complex uncertain variable is a measurable function τ 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. {τ B} {γ Γ τ(γ) B}, 3

4 Definition 8 (Peng [13]) The complex uncertainty distribution Φ(x) of a complex uncertain variable ξ is a function from C to [, 1] defined by for any complex number z. Φ(z) M{e(ξ) e(z), Im(ξ) Im(z)} In order to model complex uncertain variable, the expected value is proposed as below. Definition 9 (Peng [13]) Let ζ be a complex uncertain variable. The expected value of ζ is defined by E[ζ] E[e(ζ)] + ie[im(ζ)] provided that E[e(ζ)] and E[Im(ζ)] are finite, where E[e(ζ)] and E[Im(ζ)] are expected values of uncertain variables e(ζ) and Im(ζ), respectively. Definition 1 (Peng [13]) Suppose that ζ is a complex uncertain variable with expected value E[ζ]. Then the variance of ζ is defined by V [ζ] E[ ζ E[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. Stapulation 1 (Chen et al. [])Let ζ ξ + iτ be a complex uncertain variable with the real part ξ and imaginary part τ. The expected value of ζ exists and E[ζ] E[ξ] + ie[τ]. Assume ξ and τ are independent uncertain variables with regular uncertainty distributions Φ 1 and Φ, respectively. Then V [ζ] [(Φ 1 1 (α) E[ξ]) + (Φ 1 (α) E[τ]) ]dα. Definition 11 (Chen et al. []) 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]) ]. Stapulation (Chen et al. [])Let ζ ξ + iτ be a complex uncertain variable with the real part ξ and imaginary part τ. The expected value of ζ exists and E[ζ] E[ξ] + ie[τ]. Assume ξ and τ are independent uncertain variables with regular uncertainty distributions Φ 1 and Φ, respectively. Then V [ζ] ([(Φ 1 1 (α) E[ξ]) (Φ 1 (α) E[τ]) ] + i[(φ 1 1 (α) E[ξ])(Φ 1 (α) E[τ])])dα.. Uncertain andom Variables and Complex Uncertain andom Variables The chance space refers to the product (Γ, L, M) (Ω, A, Pr), in which (Γ, L, M) is an uncertainty space and (Ω, A, Pr) is a probability space. Definition 1 (Liu [11]) Let (Γ, L, M) (Ω, A, Pr) be a chance space, and Θ L A be an uncertain random event. Then the chance measure of Θ is defined by Ch{Θ} Pr{ω Ω M{γ Γ (γ, ω) Θ} r}dr. 4

5 It is mentioned that a chance measure satisfies normality, duality and monotonicity properties, that is (i) Ch{Γ Ω} 1; (ii) Ch{Θ} + Ch{Θ c } 1 for any event Θ; (iii) Ch{Θ 1 } Ch{Θ } for any real number set Θ 1 Θ, for more details, see Liu [11]. Besides, the subadditivity of chance measure is proved by Hou [8], that is, Ch { i1 Θ i} i1 Ch{Θ i} for a sequence of events Θ 1, Θ,. Definition 13 (Liu [11]) An uncertain random variable is a measurable function ξ from a chance space (Γ, L, M) (Ω, A, Pr) to the set of real numbers, i.e., {ξ B} is an event for any Borel set B. Theorem 3 (Liu [11]) Let ξ 1, ξ,..., ξ n be uncertain random variables on the chance space (Γ, L, M) (Ω, A, Pr) and let f be a measurable function. Then is an uncertain random variable determined by for all (γ, ω) Γ Ω. ξ f(ξ 1, ξ,..., ξ n ) ξ(γ, ω) f(ξ 1 (γ, ω), ξ (γ, ω),..., ξ n (γ, ω)) To describe an uncertain random variable, Liu [1] presented a definition of chance distribution. Definition 14 (Liu [1]) Let ξ be an uncertain random variable. Then its chance distribution is defined by for any x. Φ(x) Ch{ξ x} The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution. Theorem 4 (Liu [1]) Let η 1, η,, η m be independent random variables with probability distributions Ψ 1, Ψ,, Ψ m, respectively, and τ 1, τ,, τ n be uncertain variables. Then the uncertain random variable ξ f(η 1, η,, η m, τ 1, τ,, τ n ) has a chance distribution Φ(x) F (x, y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) m where F (x, y 1,, y m ) is the uncertainty distribution of uncertain variable f(η 1, η,, η m, τ 1, τ,, τ n ) for any real numbers y 1, y,, y m. Definition 15 (Liu [1]) Let ξ be an uncertain random variable. Then its expected value is defined by E[ξ] + provided that at least one of the two integrals is finite. Ch{ξ r}dr Ch{ξ r}dr Let Φ denote the chance distribution of ξ. Liu [1] gave a formula to calculate the expected value of uncertain random variable with chance distribution, that is, E[ξ] + (1 Φ(x))dx Φ(x)dx. 5

6 Theorem 5 (Liu [11]) Let η 1, η,, η m be independent random variables with probability distributions Ψ 1, Ψ,, Ψ m, respectively, and τ 1, τ,, τ n be independent uncertain variables (not necessarily independent). Then the uncertain random variable ξ f(η 1,, η m, τ 1,, τ n ) has an expected value E[ξ] E[f(y 1,, y m, τ 1,, τ n )]dψ 1 dψ m m where E[f(y 1,, y m, τ 1,, τ n )] is the expected value of the uncertain variable f(y 1,, y m, τ 1,, τ n ) for any real numbers y 1,, y m. Theorem 6 (Ahmadzade et al. [1]) Let η 1, η,..., η m be independent random variables with probability distributions Ψ 1, Ψ,..., Ψ m, and let τ 1, τ,..., τ n be independent uncertain variables with uncertainty distributions Υ 1, Υ,..., Υ n, respectively. Suppose ξ f(η 1,..., η m, τ 1,..., τ n ). Then E[ξ] (α, y 1,..., y m )dαdψ 1 (y 1 )...dψ m (y m ). m Definition 16 (Liu [1]) Let ξ be an uncertain random variable with a finite expected value E[ξ]. Then the variance of ξ is V [ξ] E[(ξ E[ξ]) ]. Since (ξ E[ξ]) is a nonnegative uncertain random variable, we also have V [ξ] + Ch{(ξ E[ξ]) x}dx. One question may be arisen. How to we obtain variance from chance distribution? Since the chance measure is a subadditivity measure, the variance of uncertain random variable cannot be derived simply by the chance distribution. In this case, Guo and Wang [7] suggest to a stipulation as follows: Stapulation 3 (Guo and Wang [7])Let ξ be an uncertain random variable with a chance distribution Φ. If ξ has a finite expected value E[ξ], then V [ξ] + (1 Φ ( E[ξ] + x ) + Φ ( E[ξ] x ) )dx. Based on this stipulation, Ahmadzade et al. [1] derived two formulas for calculating variance and moment of uncertain random variables via inverse uncertainty distribution. Theorem 7 (Ahmadzade et al. [1]) Let η 1, η,..., η m be independent random variables with probability distributions Ψ 1, Ψ,..., Ψ m, and let τ 1, τ,..., τ n be independent uncertain variables with uncertainty distributions Υ 1, Υ,..., Υ n, respectively. Then the variance of ξ f(η 1,..., η m, τ 1,..., τ n ) is V [ξ] (α, y 1,..., y m ) E[ξ]) dαdψ 1 (y 1 )...dψ m (y m ) m where (x, y 1,..., y m ) is the inverse uncertainty distribution of the uncertain variable f(y 1,..., y m, τ 1,..., τ n ) and is determined by Υ 1, Υ,..., Υ n. Definition 17 (Gao et al. [5]) A complex uncertain random variable is a function ξ from a chance space (Γ Ω, L A, M Pr) to the set of complex numbers such that {ξ B} {(γ, ω) Γ Ω ξ(γ, ω) B}is an event in Γ Ω for any Borel Set B of complex numbers. 6

7 Definition 18 (Gao et al. [5]) Suppose that ξ is a complex uncertain random variable. The expected value of ξ is defined by E[ξ] E[e(ξ)] + ie[im(ξ)], provided that E[e(ξ)] and E[Im(ξ)] are finite, where E[e(ξ)] and E[Im(ξ)] are expected values of uncertain random variables e(ξ) and Im(ξ), respectively. Definition 19 (Gao et al. [5]) Let ξ be a complex uncertain random variable with expected value E[ξ]. Then the variance is defined by V [ξ] E[ ξ E[ξ] ]. 3 Variance and Pseudo-Variance of Complex Uncertain andom Variables In this section, we derive a formula for calculating the expected value of a complex uncertain random variable. In addition, in order to calculate variance of a complex uncertain random variable, a stipulation is presented. For illustration of main results, several examples are explained. Theorem 8 Let η 1 and η be two independent random variables with probability distributions Ψ 1 and Ψ, respectively, and let τ 1 and τ be two independent uncertain variables with uncertainty distributions Υ 1 and Υ, respectively. Set ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then the expected value of the complex uncertain random variable ζ ξ 1 + iξ is E[ζ] [ 1 (α, y 1 ) + i (α, y )]dαdψ 1 (y 1 )dψ (y ) where i (α, y j ) is the inverse uncertainty distribution of the uncertain variable f j (y j, τ j ) and is determined by Υ j, j 1,. Proof: By invoking Definition 18 and Theorem 6, we can write E[ζ] E[ξ 1 ] + ie[ξ ] [ 1 (α, y 1 ) + i (α, y )]dαdψ 1 (y 1 )dψ (y ). Since the chance measure is a subadditivity measure, the variance of uncertain random variable cannot be derived simply by the chance distribution. In this case, we suggest to a stipulation as follows: Stapulation 4 Let η 1 and η be two independent random variables with probability distributions Ψ 1 and Ψ, respectively, and let τ 1 and τ be two independent uncertain variables with uncertainty distributions Υ 1 and Υ, respectively. Set ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then the variance of the complex uncertain random variable ζ ξ 1 + iξ is V [ζ] 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ), where j (α, y j ) is the inverse uncertainty distribution of the uncertain variable f j (y j, τ j ) and is determined by Υ j, j 1,. 7

8 Example 1 Suppose that τ 1 and τ are two uncertain variables with τ 1 L(a 1, b 1 ) and τ L(a, b ), respectively. Also, η 1 and η are two random variables with probability η 1 U(c 1, d 1 ) and η U(c, d ), respectively. Let ξ 1 τ 1 + η 1, ξ τ + η, and ζ ξ 1 + iξ. It is easy to obtain that Then the variance of ζ is V [ζ] E[ξ 1 ] a 1 + b 1 + c 1 + d 1, E[ξ ] a + b + c + d. 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ) {(y 1 + (b 1 a 1 )α + a 1 a 1 + b 1 + (y + (b a )α + a a + b + {(y 1 c 1 + d 1 {(y c + d V [τ 1 ] + V [τ ] + V [η 1 ] + V [η ] c 1 + d 1 ) c + d ) }dαdψ 1 (y 1 )dψ (y ) ) + ((b 1 a 1 )α + a 1 a 1 + b 1 )} dαdψ 1 (y 1 ) ) + ((b a )α + a a + b )} dαdψ (y ) (b i a i ) + 1 i1 (d i c i ). 1 i1 Example Suppose that τ 1 and τ are two uncertain variables with τ 1 N (e 1, σ 1 ) and τ N (e, σ ), respectively. Also, η 1 and η are two random variables with η 1 N (e 3, σ3) and η N (e 4, σ4), respectively. Let ξ 1 τ 1 + η 1, ξ τ + η, and ζ ξ 1 + iξ. It is easy to derive that E[ξ 1 ] e 1 + e 3, E[ξ ] e + e 4. It follows from Definition of expected value of random variable that (y 1 e 3 )dψ 1 (y 1 ) E[η 1 e 3 ], (y e 4 )dψ (y ) E[η e 4 ]. 8

9 Therefore, we have V [ζ] 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ) {(y 1 + e 1 + σ 1 3 α π 1 α e 1 e 3 ) + (y + e + σ 3 α π 1 α e e 4 ) }dαdψ 1 (y 1 )dψ (y ) (y 1 e 3 ) dψ 1 (y 1 ) + ( σ 3 3 α π 1 α ) dα σ 1 3 α + (y 1 e 3 )dψ 1 (y 1 ) π 1 α dα + (y e 4 ) dψ (y ) + ( σ 3 3 α π 1 α ) dα σ 3 3 α + (y e 4 )dψ (y ) π 1 α dα σ 1 + σ 3 + σ + σ 4. Example 3 Suppose that τ 1 and τ are two independent uncertain variables such that τ 1 N (e 1, σ 1 ) and τ N (e, σ ). Also, η 1 and η are two independent random variables such that η 1 N (e 3, σ 3) and η N (e 4, σ 4). Let ξ 1 τ 1 η 1, ξ τ η, and ζ ξ 1 + iξ. It is easy to obtain that E[ξ 1 ] e 1 e 3, E[ξ ] e e 4. 9

10 The definition of expected value of random variable implies that V [ζ] E[τ 1 ] 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ) {y 1 (e 1 + σ 1 3 α π 1 α ) e 1e 3 } dαdψ 1 (y 1 ) {y (e + σ 3 α π 1 α ) e e 4 } dαdψ (y ) {y 1 (e 1 + σ 1 3 α π 1 α ) y 1e 1 + y 1 e 1 e 1 e 3 }dαdψ 1 (y 1 ) {y (e + σ 3 α π 1 α )y e + y e e e 4 } dαdψ (y ) {y 1 ( σ 1 3 α π 1 α ) + e 1(y 1 e 3 )} dαdψ 1 (y 1 ) + e 1 {y ( σ 3 α π 1 α ) + e (y e 4 )} dαdψ (y ) + E[τ ] + e ( σ 1 3 α π 1 α ) dα + e 1 (y 1 e 3 ) dψ 1 (y 1 ) σ 1 3 α y 1 (y 1 e 3 )dψ 1 (y 1 ) π 1 α dα ( σ 3 α π 1 α ) dα + e 1 (y e 4 ) dψ (y ) σ 3 α y (y e 4 )dψ (y ) π 1 α dα (σ 3 + e 3)σ 1 + e 1σ 3 + (σ 4 + e 4)σ + e σ 4. Theorem 9 Let η 1 and η be two independent random variables with probability distributions Ψ 1 and Ψ, respectively, and let τ 1 and τ be two independent uncertain variables with uncertainty distributions Υ 1 and Υ, respectively. Set ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then the variance of the complex uncertain random variable ζ ξ 1 + iξ is V [ζ] [ (F 1 1 (α, y 1 )) + (α, y )) ] dαdψ 1 (y 1 )dψ (y ) E [ξ 1 ] E [ξ ], where j (α, y j ) is the inverse uncertainty distribution of the uncertain variable f j (y j, τ j ) and is determined by Υ j, j 1,. 1

11 Proof: According to Stipulation 4, we have V [ζ] + [ (F 1 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) ] dαdψ 1 (y 1 )dψ (y ) (F1 1 (α, y 1 )) dαdψ 1 (y 1 ) E[ξ 1 ] And by using Theorem 6, we have Therefore, we have V [ζ] (α, y )) dαdψ (y ) E[ξ ] E[ξ 1 ] E[ξ ] 1 (α, y 1 )dαdψ 1 (y 1 ), (α, y )dαdψ (y ). [ 1 (α, y 1 )) + (α, y )) ]dαdψ 1 (y 1 )dψ (y ) E [ξ 1 ] E [ξ ] + E [ξ 1 ] + E [ξ ] Thus the proof is completed. 1 (α, y 1 )dαdψ 1 (y 1 ) + E [ξ 1 ] (α, y )dαdψ (y ) + E [ξ 1 ]. [ 1 (α, y 1 )) + (α, y )) ]dαdψ 1 (y 1 )dψ (y ) E [ξ 1 ] E [ξ ]. In the following theorem, we obtain an upper bound for variance of sum of two complex uncertain random variables. Theorem 1 Let η 1, η, η 3, η 4 be independent random variables with probability distributions Ψ 1, Ψ, Ψ 3, Ψ 4, and let τ 1, τ, τ 3, τ 4 be independent uncertain variables with uncertainty distributions Υ 1, Υ, Υ 3, Υ 4, respectively. Set ξ 1 f 1 (η 1, τ 1 ), ξ f (η, τ ), ξ 3 f 3 (η 3, τ 3 ) and ξ 4 f 4 (η 4, τ 4 ). If the complex uncertain random variable is ζ ξ 1 + iξ, then we have V [ζ 1 + ζ ] (V [ζ 1 ] + V [ζ ]). Proof: As an application of Jensen s inequality, we have ( ) r 1 n x i 1 n x r i, r 1, n n and consequently, we obtain i1 ( n ) r x i n r 1 i1 i1 n i1 x r i, r 1. 11

12 By invoking Theorem, we have V [ζ 1 + ζ ] (α, y 1 ) + 3 (α, y ) E[ξ 1 ] E[ξ 3 ]) dαdψ 1 (y 1 )dψ 3 (y 3 ) (α, y ) + 4 (α, y 4 ) E[ξ ] E[ξ 4 ]) dαdψ (y )dψ 4 (y 4 ) [ 1 (α, y 1 ) E[ξ 1 ]) + 3 (α, y ) E[ξ 3 ])] dαdψ 1 (y 1 )dψ 3 (y 3 ) + [ (α, y ) E[ξ ]) + 4 (α, y 4 ) E[ξ 4 ])] dαdψ (y )dψ 4 (y 4 ) [ 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) ]dαdψ 1 (y 1 )dψ (y ) [ 3 (α, y 3 ) E[ξ 3 ]) + 4 (α, y 4 ) E[ξ 4 ]) ]dαdψ 3 (y 3 )dψ 4 (y 4 ) (V [ζ 1 ] + V [ζ ]). As another tool to measure degree of spread of a complex uncertain random variable, we introduce the concept of pseudo-variance of a complex uncertain random variable. Definition Suppose that ξ is a complex uncertain random variable. defined by Then the pseudo-variance is Ṽ [ξ] E[(ξ E[ξ]) ]. Since the chance measure is a subadditivity measure, the pseudo-variance of uncertain random variable cannot be derived simply by the chance distribution. In this case, we suggest to a stipulation as follows: Stapulation 5 Let η 1 and η be two independent random variables with probability distributions Ψ 1 and Ψ, respectively, and let τ 1 and τ be two independent uncertain variables with uncertainty distributions Υ 1 and Υ, respectively. Set ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then the pseudo-variance of the complex uncertain random variable ζ ξ 1 + iξ is Ṽ [ζ] ([ 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) ] + i[ 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])])dαdψ 1 (y 1 )dψ (y ) where j (α, y j ) is the inverse uncertainty distribution of the uncertain variable f j (y j, τ j ) and is determined by Υ j, j 1,. Example 4 Suppose that τ 1 and τ are two uncertain variables such that τ 1 N (e 1, σ 1 ) and τ N (e, σ ), respectively. Also, η 1 and η are two random variables such that η 1 N (e 3, σ 3) and η 1

13 N (e 4, σ 4), respectively. Let ξ 1 τ 1 +η 1, ξ τ +η, and ζ ξ 1 +iξ. It is easy to see that E[ξ 1 ] e 1 +e 3 and E[ξ ] e + e 4. It follows from Definition of expected value of random variable that (y 1 e 3 )dψ 1 (y 1 ) E[η 1 e 3 ] (y e 4 )dψ (y ) E[η e 4 ]. Then the pseudo-variance of ζ is Ṽ [ζ] {F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ) + i[ 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])]}dαdψ 1 (y 1 )dψ (y ) {(y 1 + e 1 + σ 1 3 α π 1 α e 1 e 3 ) (y + e + σ 3 α π 1 α e e 4 ) }dαdψ 1 (y 1 )dψ (y ) + i (y 1 e 3 ) dψ 1 (y 1 ) + [(y 1 e 3 ) + σ 1 3 α π 1 α ][(y e 4 ) + σ 3 α π 1 α ]dαdψ 1(y 1 )dψ (y ) ( σ 3 3 α π 1 α ) dα σ 1 3 α + (y 1 e 3 )dψ 1 (y 1 ) π 1 α dα (y e 4 ) dψ (y ) ( σ 3 3 α π 1 α ) dα σ 3 3 α (y e 4 )dψ (y ) π 1 α dα σ 3 α + i (y 1 e 3 )dψ 1 (y 1 ) π 1 α dα σ 1 3 α + i (y e 4 )dψ (y ) π 1 α dα + i (y 1 e 3 )dψ 1 (y 1 ) (y e 4 )dψ (y ) + i 3σ 1σ π ( α 1 α ) dα σ 1 + σ 3 σ σ 4 + iσ 1 σ. Theorem 11 Let η 1 and η be two independent random variables with probability distributions Ψ 1 and 13

14 Ψ, respectively, and let τ 1 and τ be two independent uncertain variables with uncertainty distributions Υ 1 and Υ, respectively. Set ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then the pseudo-variance of the complex uncertain random variable ζ ξ 1 + iξ is Ṽ [ζ] [(F1 1 (α, y 1 )) (α, y )) ]dαdψ 1 (y 1 )dψ (y ) + i 1 (α, y 1 ) (α, y )dαdψ 1 (y 1 )dψ (y ) ie[ξ 1 ]E[ξ ] where j (α, y j ) is the inverse uncertainty distribution of the uncertain variable f j (y j, τ j ) and is determined by Υ j, j 1,. Proof: It follows from Stipulation 5 that Ṽ [ζ] [(F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) ]dαdψ 1 (y 1 )dψ (y ) + i 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])dαdψ 1 (y 1 )dψ (y ) (F1 1 (α, y 1 )) dαdψ 1 (y 1 ) E[ξ 1 ] + i (α, y )) dαdψ (y ) + E[ξ ] ie[ξ ] ie[ξ 1 ] Thus the proof is finished. 1 (α, y 1 ) (α, y )dαdψ 1 (y 1 )dψ (y ) 1 (α, y 1 )dαdψ 1 (y 1 ) (α, y )dαdψ (y ) + ie[ξ 1 ]E[ξ ]. 1 (α, y 1 )dαdψ 1 (y 1 ) + E [ξ 1 ] (α, y )dαdψ (y ) + E [ξ ] In the following theorem, we prove an inequality between variance and pseudo-variance of a complex uncertain random variable. Theorem 1 Assume that η 1 and η are two independent random variables and τ 1 and τ are two independent uncertain variables. Let ξ 1 f 1 (η 1, τ 1 ) and ξ f (η, τ ). Then for the complex uncertain random variable ζ ξ 1 + iξ, we have Ṽ [ζ] V [ζ]. Proof: By invoking Stipulation 5, we have Ṽ [ζ] [(F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) ]dαdψ 1 (y 1 )dψ (y ) + i 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])dαdψ 1 (y 1 )dψ (y ). (1) 14

15 Taking norm of Equation 1, we obtain Ṽ [ζ] { [ [ + 4 [ (F 1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) ] dαdψ 1 (y 1 )dψ (y ) ] } 1 (F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])dαdψ 1 (y 1 )dψ (y ) Besides, Stipulation implies that V [ζ] (F1 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ). Therefore, we should only to show that [ [(F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ]) ]dαdψ 1 (y 1 )dψ (y )] [ + 4 [ 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])dαdψ 1 (y 1 )dψ (y ) 1 (α, y 1 ) E[ξ 1 ]) + (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ) Thus, it suffices to prove [ (F1 1 (α, y 1 ) E[ξ 1 ]) (α, y ) E[ξ ])dαdψ 1 (y 1 )dψ (y ) (F1 1 (α, y 1 ) E[ξ 1 ]) dαdψ 1 (y 1 )dψ (y ) which is concluded by Cauchy-Schwarz inequality. ] ] ] ] ] (α, y ) E[ξ ]) dαdψ 1 (y 1 )dψ (y ), Example 5 Suppose that τ 1 and τ are two uncertain variables such that τ 1 N (, 1) and τ N (, ). Also, η 1 and η are two random variables such that η 1 N (, 4) and η N (, 9). Let ξ 1 τ 1 + η 1, ξ τ + η, and ζ ξ 1 + iξ. Then Example 4 implies that Ṽ [ζ] σ1 + σ3 σ σ4 + iσ 1 σ i 8 + 4i, Ṽ [ζ] 8. Besides, by using Example, we have V [ζ] σ1 + σ3 + σ + σ It is obvious that Ṽ [ζ] 8 18 V [ζ]. 4 Conclusions In this paper, the concept of pseudo-variance of a complex uncertain random variable was proposed. By invoking the inverse uncertainty distributions of real and imaginary parts of complex uncertain random variables, we obtained two formulas for calculating the variance and pseudo-variance of a complex uncertain random variable. Furthermore, the relation between variance and pseudo-variance of a complex uncertain random variable was discussed via an inequality. 15

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