# zbMATH — the first resource for mathematics

Uncertain random variables: a mixture of uncertainty and randomness. (English) Zbl 1281.60005
The concept of uncertain random variables is proposed. This proposal is justified by the behavioral premises. The probability measure and the uncertainty measure are combined into the chance measure. The concept of chance distribution is introduced. This distribution is used for determining the expected value and variance of uncertain random variables. In my opinion, the reviewed paper presents results which are very important for applied mathematics.

##### MSC:
 60A86 Fuzzy probability
Full Text:
##### References:
 [1] Chen, XW; Liu, B, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim Decision Mak, 9, 69-81, (2010) · Zbl 1196.34005 [2] Gao, X; Gao, Y; Ralescu, DA, On liu’s inference rule for uncertain systems, Int J Uncertain Fuzziness Knowl Based Syst, 18, 1-11, (2010) · Zbl 1207.68386 [3] Kolmogorov AN (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin [4] Kruse R, Meyer KD (1987) Statistics with Vague Data. D. Reidel Publishing Company, Dordrecht · Zbl 0663.62010 [5] Kwakernaak, H, Fuzzy random variables-I: definitions and theorems, Info Sci, 15, 1-29, (1978) · Zbl 0438.60004 [6] Kwakernaak, H, Fuzzy random variables-II: algorithms and examples for the discrete case, Info Sci, 17, 253-278, (1979) · Zbl 0438.60005 [7] Liu, B, Fuzzy random chance-constrained programming, IEEE Trans Fuzzy Syst, 9, 713-720, (2001) [8] Liu, B, Fuzzy random dependent-chance programming, IEEE Trans Fuzzy Syst, 9, 721-726, (2001) [9] Liu, B, Random fuzzy dependent-chance programming and its hybrid intelligent algorithm, Info Sci, 141, 259-271, (2002) · Zbl 1175.90439 [10] Liu, B; Liu, YK, Expected value of fuzzy variable and fuzzy expected value models, IEEE Trans Fuzzy Syst, 10, 445-450, (2002) [11] Liu B (2004) Uncertainty theory: an Introduction to its Axiomatic Foundations. Springer, Berlin · Zbl 1072.28012 [12] Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin · Zbl 1141.28001 [13] Liu, B, Fuzzy process, hybrid process and uncertain process, J Uncertain Syst, 2, 3-16, (2008) [14] Liu, B, Some research problems in uncertainty theory, J Uncertain Syst, 3, 3-10, (2009) [15] Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin · Zbl 1158.90010 [16] Liu, B, Uncertain set theory and uncertain inference rule with application to uncertain control, J Uncertain Syst, 4, 83-98, (2010) [17] Liu, B, Uncertain risk analysis and uncertain reliability analysis, J Uncertain Syst, 4, 163-170, (2010) [18] Liu B (2010c) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin · Zbl 1225.93121 [19] Liu, B, Uncertain logic for modeling human language, J Uncertain Syst, 5, 3-20, (2011) [20] Liu, B, Why is there a need for uncertainty theory, J Uncertain Syst, 6, 3-10, (2012) [21] Liu B (2012b) Membership functions and operational law of uncertain sets. Fuzzy Optim Decision Mak (in press) [22] Liu, YH; Ha, MH, Expected value of function of uncertain variables, J Uncertain Syst, 4, 181-186, (2010) [23] Liu YH, Chen XW (2009) Uncertain currency model and currency option pricing, http://orsc.edu.cn/online/091010.pdf [24] Liu, YK; Liu, B, Fuzzy random variables: a scalar expected value operator, Fuzzy Optim Decision Mak, 2, 143-160, (2003) [25] Liu, YK; Liu, B, Fuzzy random programming with equilibrium chance constraints, Infor Sci, 170, 363-395, (2005) · Zbl 1140.90520 [26] Peng, J; Yao, K, A new option pricing model for stocks in uncertainty markets, Int J Oper Res, 8, 18-26, (2011) [27] Peng, ZX; Iwamura, K, A sufficient and necessary condition of uncertainty distribution, J Interdiscip Math, 13, 277-285, (2010) · Zbl 1229.28029 [28] Puri, ML; Ralescu, DA, Fuzzy random variables, J Math Anal Appl, 114, 409-422, (1986) · Zbl 0592.60004 [29] Yao K (2012) Uncertain calculus with renewal process. Fuzzy Optim Decision Mak (in press) · Zbl 1225.93121 [30] Zadeh, LA, Fuzzy sets, Info Control, 8, 338-353, (1965) · Zbl 0139.24606 [31] Zadeh, LA, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst, 1, 3-28, (1978) · Zbl 0377.04002 [32] Zhu, Y, Uncertain optimal control with application to a portfolio selection model, Cybern Syst, 41, 535-547, (2010) · Zbl 1225.93121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.