New independence definition of fuzzy random variable and random fuzzy variable

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1 ISSN , England, UK World Journal of Modelling and Simulation Vol. 2 (2006) No. 5, pp New independence definition of fuzzy random variable and random fuzzy variable Xiang Li, Baoding Liu Department of Mathematical Sciences, Tsinghua University, Beijing , China (Received October , Accepted December ) Abstract.Fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables, while random fuzzy variable is a function from a credibility space to the set of random variables. The concepts of independent and identically distributed fuzzy random variables and random fuzzy variables are presented, and some useful properties are discussed. Keywords: fuzzy variable, fuzzy random variable, random fuzzy variable, independence 1 Introduction The concept of fuzzy random variable was introduced by Kwakernaak [5, 6] as a natural generalization of random variable in order to represent relationships between the outcomes of a random experiment and nonstatistical inexact data. Generally speaking, fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables. According to different requirements of measurability, different researchers have studied fuzzy random variable such as Kruse and Meyer [4], Negoita and Ralescu [13], Puri and Ralescu [14], and Liu and Liu [12]. The concept of chance measure of fuzzy random event was first given by Liu [7, 8]. In order to rank fuzzy random variables, Liu and Liu [12] presented a scalar expected value operator. In addition, Liu and Liu [12] introduced the concept of independent and identically distributed fuzzy random variables. In order to describe fuzzy random variable, Yang and Liu [16] presented the concept of chance distribution. In 2002, Liu [9] defined a new concept of random fuzzy variable as a function from a credibility space to the set of random variables. The concept of chance measure of random fuzzy event was first given by Liu [9]. In order to rank random fuzzy variables, Liu and Liu [3] presented a scalar expected value operator. In addition, Liu [10] introduced the concept of independent and identically distributed random fuzzy variables. In order to describe random fuzzy variable, Zhu and Liu [18] presented the concept of chance distribution. The purpose of this paper is to present new definitions of independent and identically distributed fuzzy random variables, independent and identically distributed random fuzzy variables. This paper is organized as follows. Section 2 recalls some definitions and properties of fuzzy variable which are useful in the rest of this paper. Section 3 introduces the concept of independent and identically distributed fuzzy random variables. In Section 4, the concept of independent and identically distributed random fuzzy variables is given. At the end of this paper, a brief summary is given. 2 Fuzzy variables Let ξ be a fuzzy variable with membership function µ. In order to deal with fuzzy event, Liu and Liu [11] gave the concept of credibility measure by This work is supported by National Natural Science Foundation of China (No ). address: liu@tsinghua.edu.cn Published by World Academic Press, World Academic Union

2 World Journal of Modelling and Simulation, Vol. 2 (2006) No. 5, pp Cr{ξ B} = 1 ( ) sup µ(x) + 1 sup µ(x) 2 x B x B c for any set B of real numbers. Conversely, if ξ is a fuzzy variable, then its membership function is derived from the credibility measure by µ(x) = (2Cr{ξ = x}) 1, x R. The vector (ξ 1, ξ 2,, ξ n ) is called an n-dimensional fuzzy vector if ξ i are fuzzy variables for all i. The independence of fuzzy variables has been discussed by many authors from different angles, for example, Zadeh [17], Yager [15], Liu [10], Liu and Gao [2]. A lot of equivalence conditions of independence are presented. Here we use the condition given by Liu and Gao [2]. That is, the fuzzy variables ξ 1, ξ 2,, ξ n are said to be independent if and only if } { n Cr {ξ i B i } i=1 = min 1 i n Cr{ξ i B i } for any sets B 1, B 2,, B n of R. Furthermore, if ξ 1, ξ 2,, ξ n are independent fuzzy variables, and f i : R R functions for i = 1, 2,, n. Then f 1 (ξ 1 ), f 2 (ξ 2 ),, f n (ξ n ) are independent fuzzy variables. The concept of identically distributed fuzzy variables was given by Liu [10]. That is, the fuzzy variables ξ and η are said to be identically distributed if Cr{ξ B} = Cr{η B} for any set B of R. Furthermore, if ξ and η are identically distributed fuzzy variables, and f : R R a function. Then f(ξ) and f(η) are identically distributed fuzzy variables. Liu and Liu [11] presented the expected value of a fuzzy variable ξ as follows, E[ξ] = + 0 Cr{ξ r}dr provided that at least one of the two integrals is finite. 0 Cr{ξ r}dr 3 Fuzzy random variables According to different requirements of measurability, different definitions of fuzzy random variable were given such as Kruse and Meyer [4], Negoita and Ralescu [13], Puri and Ralescu [14]. For our purpose, we use the following definition given by Liu and Liu [12]. Definition 1. A fuzzy random variable is a function from a probability space (Ω, A, Pr) to the set of fuzzy variables such that Cr{ξ(ω) B} is a measurable function of ω for any Borel set B of R. The following properties of fuzzy random variable were given by Liu and Liu [12]. If the expected value E[ξ(ω)] is finite for each ω, then E[ξ( )] is a random variable. Furthermore, suppose that ξ is a fuzzy random variable, f : R R a measurable function. Then f(ξ) is a fuzzy random variable. The chance of fuzzy random event was defined by Liu [10], Gao and Liu [1] as Ch{ξ B}(α) = sup inf P r{a} α ω A Cr{ξ(ω) B}. The chance distribution Φ(x; α) : R (0, 1] [0, 1] is defined by Yang and Liu [15] as Φ(x; α) = Ch{ξ x}(α). The concept of independent and identically distributed fuzzy random variables have been defined by Liu and Liu [12] via independent and identically distributed random vector. For simplicity, we introduce the following new definition. Definition 2. Fuzzy random variables ξ 1, ξ 2,, ξ n are said to be independent if (a) ξ 1 (ω), ξ 2 (ω),, ξ n (ω) are independent fuzzy variables for each ω; (b) E[ξ 1 ( )], E[ξ 2 ( )],, E[ξ n ( )] are independent random variables. WJMS for subscription: info@wjms.org.uk

3 340 X. Li & B. Liu: New independence definition of fuzzy random variable Theorem 1. If ξ 1, ξ 2,, ξ n are independent fuzzy random variables, then a 1 ξ 1 + b 1, a 2 ξ 2 + b 2,, a n ξ n + b n are independent for any a i, b i of R, i = 1, 2,, n. Proof: Since ξ 1, ξ 2,, ξ n are independent, it follows from Definition 3.2 (a) that ξ 1 (ω), ξ 2 (ω),, ξ n (ω) are independent fuzzy variables for each ω. Thus fuzzy variables a 1 ξ 1 (ω)+b 1, a 2 ξ 2 (ω)+b 2,, a n ξ n (ω)+b 2 are independent for each ω. It follows from Definition 3.2 (b) that E[ξ 1 (ω)], E[ξ 2 (ω)],, E[ξ n (ω)] are independent random variables. Thus we have E[a i ξ i (ω) + b i ] = a i E[ξ i (ω)] + b i, i = 1, 2,, n are independent random variables. It follows from Definition 3.2 that fuzzy random variables a 1 ξ 1 + b 1, a 2 ξ 2 + b 2,, a n ξ n + b n are independent for any a i, b i of R, i = 1, 2,, n. The proof is complete. Definition 3. The fuzzy random variables ξ and η are said to be identically distributed if for any α (0, 1] and Borel set B of R. Ch{ξ B}(α) = Ch{η B}(α) Theorem 2. Let ξ and η be identically distributed fuzzy random variables, and f : R R a measurable function. Then f(ξ) and f(η) are identically distributed fuzzy random variables. Proof: For any α (0, 1] and Borel set B of R, we have Ch{f(ξ) B}(α) = Ch{ξ f 1 (B)}(α) = Ch{η f 1 (B)}(α) = Ch{f(η) B}(α). It follows from Definition 3.3 that f(ξ) and f(η) are identically distributed fuzzy random variables. The proof is complete. Theorem 3. If ξ and η are identically distributed fuzzy random variables, then ξ and η have the same chance distribution. Proof: If ξ and η are identically distributed fuzzy random variables, then, for any α (0, 1] and x R, we have Ch{ξ (, x]}(α) = Ch{η (, x]}(α). Thus ξ and η have the same chance distribution. 4 Random fuzzy variables Random fuzzy variable was first introduced by Liu [9] as a fuzzy variable taking random variable values. Formally, we have the following definition. Definition 4. A random fuzzy variable is a function from a credibility space (Θ, P (Θ), Cr) to the set of random variables. The following properties of random fuzzy variable were given by Liu and Liu [9]. Let ξ be a random fuzzy variable. If the expected value E[ξ(θ)] is finite for each θ, then E[ξ( )] is a fuzzy variable. Furthermore, suppose f : R R is a measurable function, then f(ξ) is a random fuzzy variable. The chance of random fuzzy event was defined by Liu [9] as Ch{ξ B}(α) = sup Cr{A} α θ A inf P r{ξ(θ) B}. Then the chance distribution Φ(x; α) : R (0, 1] [0, 1] is defined by Zhu and Liu [18] as Φ(x; α) = Ch{ξ x}(α). The concept of independent and identically distributed random fuzzy variables has been defined by Liu [10] via independent and identically distributed fuzzy vector. For simplicity, we introduce the following new definition. WJMS for contribution: submit@wjms.org.uk

4 World Journal of Modelling and Simulation, Vol. 2 (2006) No. 5, pp Definition 5. Random fuzzy variables ξ 1, ξ 2,, ξ n are said to be independent if (a) ξ 1 (θ), ξ 2 (θ),, ξ n (θ) are independent random variables for each θ; (b) E[ξ 1 ( )], E[ξ 2 ( )],, E[ξ n ( )] are independent fuzzy variables. Theorem 4. If ξ 1, ξ 2,, ξ n are independent random fuzzy variables, then a 1 ξ 1 + b 1, a 2 ξ 2 + b 2,, a n ξ n + b n are independent for any a i, b i of R, i = 1, 2,, n. Proof: The proof is similar to Theorem 3.1. Definition 6. The random fuzzy variables ξ and η are said to be identically distributed if for any α (0, 1] and Borel set B of R. Ch{ξ B}(α) = Ch{η B}(α) Theorem 5. Let ξ and η be identically distributed random fuzzy variables, and f : R R a measurable function. Then f(ξ) and f(η) are identically distributed random fuzzy variables. Proof: The proof is similar to Theorem 3.2. Theorem 6. If ξ and η are identically distributed random fuzzy variables, then ξ and η have the same chance distribution. Proof: The proof is similar to Theorem Conclusions This paper contributes to the research area of fuzzy random theory and random fuzzy theory in the following aspects: (a) the concept of independent and identically distributed fuzzy random variables was presented; and (b) the concept of independent and identically distributed random fuzzy variables was presented. References [1] J. Gao, B. Liu. New primitive chance measures of fuzzy random event. International Journal of Fuzzy, 2001, 3(4): [2] Y. K.Liu, J. Gao. The independence of fuzzy variables in credinility theory and its applications. Technical Report, [3] Y. K.Liu, B. Liu. Expected value operator of random fuzzy variable and random fuzzy expected value models. International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, 2003, 11: [4] R. Kruse, K. D. Meyer. Statistics with Vague Data. D. Reidel Publishing Company, Dordrecht. [5] H. Kwakernaak. Fuzzy random variables-i. definitions and theorems. Information Sciences, 1978, 15: [6] H. Kwakernaak. Fuzzy random variables-ii. algorithms and examples for the discrete case. Information Sciences, 1979, 17: [7] B. Liu. Fuzzy random chance-constrained programming. IEEE Transactions on Fuzzy Systems, 2001, 9(5): [8] B. Liu. Fuzzy random dependent-chance programming. IEEE Transactions on Fuzzy Systems, 2001, 9(5): [9] B. Liu. Theory and Practice of Uncertain Programming. Physica-Verlag, Heidelberg, [10] B. Liu. Uncertainty Theory: An Introduction to its Axiomatic Foundations. Springer-Verlag, Berlin, [11] B. Liu, Y. K. Liu. Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 2002, 10(4): [12] Y. K. Liu, B. Liu. Fuzzy random variables: A scalar expected value operator. Fuzzy Optimization and Decision Making, 2003, 2(2): [13] C. V. Negoita, D. Ralescu. Simulation, Knowledge-Based Computing and Fuzzy Statistcs. Van Nostrand Reinhold Company, New York, [14] M. L. Puri, D. Ralescu. Fuzzy random variables. Journal of Mathematical Analysis and Applications, 1986, 114: WJMS for subscription: info@wjms.org.uk

5 342 X. Li & B. Liu: New independence definition of fuzzy random variable [15] R. R. Yager. On the specificity of a possibility distribution. Fuzzy Sets and Systems, 1992, 50: [16] L. Yang, B. Liu. Chance distribution of fuzzy random variable and laws of large numbers. Technical Report, [17] L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1978, 1: [18] Y. Zhu, B. Liu. Continuity theorems and chance distribution of random fuzzy variable. Proceedings of the Royal Society of London Series A, 2004, 460: WJMS for contribution: submit@wjms.org.uk