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Fuzzy Probability Theory

  • Reference work entry
Computational Complexity

  • 314 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Mathematical Environment

Fuzzy Random Quantities

Fuzzy Probability

Representation of Fuzzy Random Quantities

Future Directions

Bibliography

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Abbreviations

Fuzzy set and fuzzy vector :

Let \( { \underline{\mathbf{X}} } \) represent a universal set and \( { \underline{x} } \) be the elements of \( { \underline{\mathbf{X}} } \), then

$$ \tilde{\underline{A}}=\{(\underline{x}, \mu_{A}(\underline{x}))\,|\, \underline{x}\in\underline{\mathbf{X}}\},\: \mu_{A}(\underline{x})\ge0\quad\forall \underline{x}\in\underline{\mathbf{X}} $$
(1)

is referred to as fuzzy set \( { \tilde{\underline{A}} } \) on \( { \underline{\mathbf{X}} } \).\( { \mu_{A}(\underline{x}) } \) is the membership function (characteristic function) of the fuzzy set \( { \tilde{\underline{A}} } \) and represents the degree with which the elements \( { \underline{x} } \) belong to \( { \tilde{\underline{A}} } \). If

$$ \sup_{\underline{x}\in \underline{\mathbf{X}}}[\mu_{A} (\underline{x})]=1\:, $$
(2)

the membership function and the fuzzy set \( { \tilde{\underline{A}} } \) are called normalized; see Fig. 1. In case of a limitation to the Euclidean space \( { \underline{\mathbf{X}}=\mathbb{R}^{n} } \) and normalized fuzzy sets, the fuzzy set \( { \tilde{\underline{A}} } \) is also referred to as fuzzy vector denoted by \( { \underline{\tilde{x}} } \) with its membership function \( { \mu (\underline{x}) } \), or, in the one‐dimensional case, as fuzzy variable \( { \tilde{x} } \) with \( { \mu (x) } \).

α‑Level set and support:

The crisp sets

$$ \underline{A}_{\alpha_{k}}= \{\underline{x}\in\underline{\mathbf{X}}\,|\, \mu_{A}(\underline{x})\ge\alpha_{k}\} $$
(3)

extracted from the fuzzy set \( { \tilde{\underline{A}} } \) for real numbers \( { \alpha_{k}\in(0,1] } \) are called α‑level sets.These comply with the inclusion property

$$ \underline{A}_{\alpha_{k}}\subseteq\underline{A}_{\alpha_{i}} \quad\forall\alpha_{i},\alpha_{k}\in(0,1] \text{ with } \alpha_{i}\leq\alpha_{k}\:.$$
(4)

The largest α‑level set \( { \underline{A}_{\alpha_{k}\to +0} } \) is called support \( { S(\underline{\tilde{A}}) } \); see Fig. 1.

Figure 1
figure 1

Normalized fuzzy set with α‑level sets and support

Full size image
σ‑Algebra:

A family \( { \mathfrak{M}(\underline{\mathbf{X}}) } \) of sets \( { \underline{A}_{i} } \) on the universal set \( { \underline{\mathbf{X}} } \) is referred to as σ‑algebra \( { \mathfrak{S}(\underline{\mathbf{X}}) } \) on \( { \underline{\mathbf{X}} } \), if

$$ \underline{\mathbf{X}}\in \mathfrak{S}(\underline{\mathbf{X}})\:, $$
(5)
$$ \underline{A}_{i}\in\mathfrak{S} (\underline{\mathbf{X}}) \Rightarrow \underline{A}_{i}^{C}\in\mathfrak{S} (\underline{\mathbf{X}})\:, $$
(6)

and if for every sequence of sets \( { \underline{A}_{i} } \)

$$ \underline{A}_{i}\in\mathfrak{S}(\underline{\mathbf{X}});\, i=1,2,\ldots\Rightarrow\bigcup_{i}^{\infty}\underline{A}_{i}\in\mathfrak{S} (\underline{\mathbf{X}})\:.$$
(7)

In this definition, \( { \underline{A}_{i}^{C} } \) is the complementary set of \( { \underline{A}_{i} } \) with respect to \( { \underline{\mathbf{X}} } \), a family \( { \mathfrak{M}(\underline{\mathbf{X}}) } \) of sets \( { \underline{A}_{i} } \) refers to subsets and systems of subsets of the power set \( { \mathfrak{P}(\underline{\mathbf{X}}) } \) on \( { \underline{\mathbf{X}} } \), and the power set \( { \mathfrak{P}(\underline{\mathbf{X}}) } \) is the set of all subsets \( { \underline{A}_{i} } \) of \( { \underline{\mathbf{X}} } \).

Bibliography

Primary Literature

  1. Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New York

    MATH  Google Scholar 

  2. Bandemer H (1992) Modelling uncertain data.Akademie-Verlag, Berlin

    MATH  Google Scholar 

  3. Bandemer H, Gebhardt A (2000) Bayesian fuzzy kriging.Fuzzy Sets Syst 112:405–418

    Article  MathSciNet  MATH  Google Scholar 

  4. Bandemer H, Gottwald S (1995) Fuzzy sets, fuzzy logic fuzzy methods with applications.Wiley, Chichester

    MATH  Google Scholar 

  5. Bandemer H, Näther W (1992) Fuzzy data analysis.Kluwer, Dordrecht

    Google Scholar 

  6. Beer M (2007) Model-free sampling.Struct Saf 29:49–65

    Article  Google Scholar 

  7. Berleant D, Zhang J (2004) Representation and problem solving with distribution envelope determination (denv).Reliab Eng Syst Saf 85(1–3):153–168

    Article  Google Scholar 

  8. Bernardini A, Tonon F (2004) Aggregation of evidence from random and fuzzy sets.Special Issue of ZAMM.Z Angew Math Mech 84(10–11):700–709

    Article  MathSciNet  MATH  Google Scholar 

  9. Bodjanova S (2000) A generalized histogram.Fuzzy Sets Syst 116:155–166

    Article  MATH  Google Scholar 

  10. Colubi A, Domínguez-Menchero JS, López-Díaz M, Gil MA (1999) A generalized strong law of large numbers.Probab Theor Relat Fields 14:401–417

    Google Scholar 

  11. Colubi A, Domínguez-Menchero JS, López-Díaz M, Ralescu DA (2001) On the formalization of fuzzy random variables.Inf Sci 133:3–6

    Google Scholar 

  12. Colubi A, Domínguez-Menchero JS, López-Díaz M, Ralescu DA (2002) A de[0, 1]-representation of random upper semicontinuous functions.Proc Am Math Soc 130:3237–3242

    Google Scholar 

  13. Colubi A, Fernández-García C, Gil MA (2002) Simulation of random fuzzy variables: an empirical approach to statistical/probabilistic studies with fuzzy experimental data.IEEE Trans Fuzzy Syst 10:384–390

    Google Scholar 

  14. Couso I, Sanchez L (2008) Higher order models for fuzzy random variables. Fuzzy Sets Syst 159:237–258

    Article  MathSciNet  MATH  Google Scholar 

  15. de Cooman G (2002) The society for imprecise probability: theories and applications.http://www.sipta.org

  16. Diamond P (1990) Least squares fitting of compact set-valued data.J Math Anal Appl 147:351–362

    Article  MathSciNet  MATH  Google Scholar 

  17. Diamond P, Kloeden PE (1994) Metric spaces of fuzzy sets: theory and applications.World Scientific, Singapore

    Book  MATH  Google Scholar 

  18. Dubois D, Prade H (1980) Fuzzy sets and systems theory and applications. Academic Press, New York

    MATH  Google Scholar 

  19. Dubois D, Prade H (1985) A review of fuzzy set aggregation connectives.Inf Sci 36:85–121

    Article  MathSciNet  MATH  Google Scholar 

  20. Dubois D, Prade H (1986) Possibility theory.Plenum Press, New York

    Google Scholar 

  21. Fellin W, Lessmann H, Oberguggenberger M, Vieider R (eds) (2005) Analyzing uncertainty in civil engineering.Springer, Berlin

    Google Scholar 

  22. Feng Y, Hu L, Shu H (2001) The variance and covariance of fuzzy random variables and their applications.Fuzzy Sets Syst 120(3):487–497

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferson S, Hajagos JG (2004) Arithmetic with uncertain numbers: rigorous and (often) best possible answers.Reliab Eng Syst Saf 85(1–3):135–152

    Article  Google Scholar 

  24. Fetz T, Oberguggenberger M (2004) Propagation of uncertainty through multivariate functions in the framework of sets of probability measures.Reliab Eng Syst Saf 85(1–3):73–87

    Article  Google Scholar 

  25. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York; Revised edition 2003, Dover Publications, Mineola

    Book  Google Scholar 

  26. González-Rodríguez G, Montenegro M, Colubi A, Ángeles Gil M (2006) Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data.Fuzzy Sets Syst 157(19):2608–2613

    Google Scholar 

  27. Grzegorzewski P (2000) Testing statistical hypotheses with vague data. Fuzzy Sets Syst 112:501–510

    Article  MathSciNet  MATH  Google Scholar 

  28. Hall JW, Lawry J (2004) Generation, combination and extension of random set approximations to coherent lower and upper probabilities.Reliab Eng Syst Saf 85(1–3):89–101

    Article  Google Scholar 

  29. Helton JC, Johnson JD, Oberkampf WL (2004) An exploration of alternative approaches to the representation of uncertainty in model predictions.Reliab Eng Syst Saf 85(1–3):39–71

    Article  Google Scholar 

  30. Helton JC, Oberkampf WL (eds) (2004) Special issue on alternative representations of epistemic uncertainty. Reliab Eng Syst Saf 85(1–3):1–369

    Google Scholar 

  31. Hung W-L (2001) Bootstrap method for some estimators based on fuzzy data. Fuzzy Sets Syst 119:337–341

    Article  Google Scholar 

  32. Hwang C-M, Yao J-S (1996) Independent fuzzy random variables and their application.Fuzzy Sets Syst 82:335–350

    Article  MathSciNet  MATH  Google Scholar 

  33. Jang L-C, Kwon J-S (1998) A uniform strong law of large numbers for partial sum processes of fuzzy random variables indexed by sets.Fuzzy Sets Syst 99:97–103

    Article  MathSciNet  MATH  Google Scholar 

  34. Joo SY, Kim YK (2001) Kolmogorovs strong law of large numbers for fuzzy random variables.Fuzzy Sets Syst 120:499–503

    Article  MathSciNet  MATH  Google Scholar 

  35. Kim YK (2002) Measurability for fuzzy valued functions.Fuzzy Sets Syst 129:105–109

    Article  MATH  Google Scholar 

  36. Klement EP, Puri ML, Ralescu DA (1986) Limit theorems for fuzzy random variables.Proc Royal Soc A Math Phys Eng Sci 407:171–182

    Article  MathSciNet  MATH  Google Scholar 

  37. Klement EP (1991) Fuzzy random variables.Ann Univ Sci Budapest Sect Comp 12:143–149

    MathSciNet  MATH  Google Scholar 

  38. Klir GJ (2006) Uncertainty and information: foundations of generalized information theory.Wiley-Interscience, Hoboken

    Google Scholar 

  39. Klir GJ, Folger TA (1988) Fuzzy sets, uncertainty, and information. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  40. Körner R (1997) Linear models with random fuzzy variables.Phd thesis, Bergakademie Freiberg, Fakultät für Mathematik und Informatik

    Google Scholar 

  41. Körner R (1997) On the variance of fuzzy random variables.Fuzzy Sets Syst 92:83–93

    Google Scholar 

  42. Körner R, Näther W (1998) Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates.Inf Sci 109:95–118

    Google Scholar 

  43. Krätschmer V (2001) A unified approach to fuzzy random variables.Fuzzy Sets Syst 123:1–9

    Google Scholar 

  44. Krätschmer V (2002) Limit theorems for fuzzy-random variables.Fuzzy Sets Syst 126:253–263

    Google Scholar 

  45. Krätschmer V (2004) Probability theory in fuzzy sample space.Metrika 60:167–189

    Google Scholar 

  46. Kruse R, Meyer KD (1987) Statistics with vague data.Reidel, Dordrecht

    Book  MATH  Google Scholar 

  47. Kwakernaak H (1978) Fuzzy random variables I. definitions and theorems.Inf Sci 15:1–19

    Article  MathSciNet  MATH  Google Scholar 

  48. Kwakernaak H (1979) Fuzzy random variables II. algorithms and examples for the discrete case.Inf Sci 17:253–278

    Article  MathSciNet  MATH  Google Scholar 

  49. Li S, Ogura Y, Kreinovich V (2002) Limit theorems and applications of set valued and fuzzy valued random variables.Kluwer, Dordrecht

    Book  Google Scholar 

  50. Lin TY, Yao YY, Zadeh LA (eds) (2002) Data mining, rough sets and granular computing.Physica, Germany

    MATH  Google Scholar 

  51. López-Díaz M, Gil MA (1998) Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications.J Stat Plan Inference 74:11–29

    Google Scholar 

  52. Matheron G (1975) Random sets and integral geometry.Wiley, New York

    MATH  Google Scholar 

  53. Möller B, Graf W, Beer M (2000) Fuzzy structural analysis using alpha-level optimization.Comput Mech 26:547–565

    Google Scholar 

  54. Möller B, Beer M (2004) Fuzzy randomness – uncertainty in civil engineering and computational mechanics.Springer, Berlin

    Google Scholar 

  55. Möller B, Reuter U (2007) Uncertainty forecasting in engineering.Springer, Berlin

    Google Scholar 

  56. Muhanna RL, Mullen RL, Zhang H (2007) Interval finite element as a basis for generalized models of uncertainty in engineering mechanics.J Reliab Comput 13(2):173–194

    Article  MathSciNet  MATH  Google Scholar 

  57. Gil MA, López-Díaz M, Ralescu DA (2006) Overview on the development of fuzzy random variables.Fuzzy Sets Syst 157(19):2546–2557

    Google Scholar 

  58. Näther W, Körner R (2002) Statistical modelling, analysis and management of fuzzy data, chapter on the variance of random fuzzy variables.Physica, Heidelberg, pp 25–42

    Google Scholar 

  59. Näther W (2006) Regression with fuzzy random data.Comput Stat Data Analysis 51:235–252

    Google Scholar 

  60. Näther W, Wünsche A (2007) On the conditional variance of fuzzy random variables.Metrika 65:109–122

    Google Scholar 

  61. Oberkampf WL, Helton JC, Sentz K (2001) Mathematical representation of uncertainty.In: AIAA non-deterministic approaches forum, number AIAA 2001–1645. AIAA, Seattle

    Google Scholar 

  62. Pedrycz W, Skowron A, Kreinovich V (eds) (2008) Handbook of granular computing.Wiley, New York

    Google Scholar 

  63. Puri ML, Ralescu D (1986) Fuzzy random variables.J Math Anal Appl 114:409–422

    Article  MathSciNet  MATH  Google Scholar 

  64. Puri ML, Ralescu DA (1983) Differentials of fuzzy functions.J Math Anal Appl 91:552–558

    Article  MathSciNet  MATH  Google Scholar 

  65. Puri ML, Ralescu DA (1991) Convergence theorem for fuzzy martingales.J Math Anal Appl 160:107–122

    Article  MathSciNet  MATH  Google Scholar 

  66. Rodríguez-Muñiz L, López-Díaz M, Gil MA (2005) Solving influence diagrams with fuzzy chance and value nodes.Eur J Oper Res 167:444–460

    Google Scholar 

  67. Samarasooriya VNS, Varshney PK (2000) A fuzzy modeling approach to decision fusion under uncertainty.Fuzzy Sets Syst 114:59–69

    Article  MATH  Google Scholar 

  68. Schenk CA, Schuëller GI (2005) Uncertainty assessment of large finite element systems.Springer, Berlin

    Google Scholar 

  69. Schuëller GI, Spanos PD (eds) (2001) Proc int conf on monte carlo simulation MCS 2000.Swets and Zeitlinger, Monaco

    Google Scholar 

  70. Shafer G (1976) A mathematical theory of evidence.Princeton University Press, Princeton

    MATH  Google Scholar 

  71. Song Q, Leland RP, Chissom BS (1997) Fuzzy stochastic fuzzy time series and its models.Fuzzy Sets Syst 88:333–341

    Article  MathSciNet  MATH  Google Scholar 

  72. Taheri SM, Behboodian J (2001) A bayesian approach to fuzzy hypotheses testing.Fuzzy Sets Syst 123:39–48

    Article  MathSciNet  MATH  Google Scholar 

  73. Terán P (2006) On borel measurability and large deviations for fuzzy random variables.Fuzzy Sets Syst 157(19):2558–2568

    Google Scholar 

  74. Terán P (2007) Probabilistic foundations for measurement modelling with fuzzy random variables.Fuzzy Sets Syst 158(9):973–986

    Google Scholar 

  75. Viertl R (1996) Statistical methods for non-precise data.CRC Press, Boca Raton

    Google Scholar 

  76. Viertl R, Hareter D (2004) Generalized Bayes' theorem for non-precise a-priori distribution.Metrika 59:263–273

    Article  MathSciNet  MATH  Google Scholar 

  77. Viertl R, Trutschnig W (2006) Fuzzy histograms and fuzzy probability distributions. In: Proceedings of the 11th Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems, Editions EDK, Paris, CD-ROM

    Google Scholar 

  78. Walley P (1991) Statistical reasoning with imprecise probabilities.Chapman & Hall, London

    MATH  Google Scholar 

  79. Wang G, Qiao Z (1994) Convergence of sequences of fuzzy random variables and its application.Fuzzy Sets Syst 63:187–199

    Article  MathSciNet  MATH  Google Scholar 

  80. Wang G, Zhang Y (1992) The theory of fuzzy stochastic processes.Fuzzy Sets Syst 51:161–178

    Article  MATH  Google Scholar 

  81. Weichselberger K (2000) The theory of interval-probability as a unifying concept for uncertainty.Int J Approx Reason 24(2–3):149–170

    Article  MathSciNet  MATH  Google Scholar 

  82. Yager RR (1984) A representation of the probability of fuzzy subsets.Fuzzy Sets Syst 13:273–283

    Article  MathSciNet  MATH  Google Scholar 

  83. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  84. Zadeh LA (1968) Probability measures of fuzzy events.J Math Anal Appl 23:421–427

    Article  MathSciNet  MATH  Google Scholar 

  85. Zhong Q, Yue Z, Guangyuan W (1994) On fuzzy random linear programming. Fuzzy Sets Syst 65(1):31–49

    Article  MathSciNet  MATH  Google Scholar 

  86. Zimmermann HJ (1992) Fuzzy set theory and its applications.Kluwer, Boston

    Google Scholar 

Books and Reviews

  1. Ayyub BM (1998) Uncertainty modeling and analysis in civil engineering.CRC Press, Boston

    Google Scholar 

  2. Blockley DI (1980) The nature of structural design and safety.Ellis Horwood, Chichester

    Google Scholar 

  3. Buckley JJ (2003) Fuzzy Probabilities. Physica/Springer, Heidelberg

    Book  MATH  Google Scholar 

  4. Cai K-Y (1996) Introduction to fuzzy reliability.Kluwer, Boston

    Book  MATH  Google Scholar 

  5. Chou KC, Yuan J (1993) Fuzzy-bayesian approach to reliability of existing structures.ASCE J Struct Eng 119(11):3276–3290

    Article  Google Scholar 

  6. Elishakoff I (1999) Whys and hows in uncertainty modelling probability, fuzziness and anti-optimization.Springer, New York

    Google Scholar 

  7. Feng Y (2000) Decomposition theorems for fuzzy supermartingales and submartingales.Fuzzy Sets Syst 116:225–235

    Article  MATH  Google Scholar 

  8. Gil MA, López-Díaz M (1996) Fundamentals and bayesian analyses of decision problems with fuzzy-valued utilities.Int J Approx Reason 15:203–224

    Google Scholar 

  9. Gil MA, Montenegro M, González-Rodríguez G, Colubi A, Casals MR (2006) Bootstrap approach to the multi-sample test of means with imprecise data.Comput Stat Data Analysis 51:148–162

    Google Scholar 

  10. Grzegorzewski P (2001) Fuzzy sets b defuzzification and randomization. Fuzzy Sets Syst 118:437–446

    Article  MathSciNet  MATH  Google Scholar 

  11. Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric.Fuzzy Sets Syst 148(2):319–328

    Article  MathSciNet  MATH  Google Scholar 

  12. Hareter D (2004) Time series analysis with non-precise data. In: Wojtkiewicz S, Red-Horse J, Ghanem R (eds) 9th ASCE specialty conference on probabilistic mechanics and structural reliability.Sandia National Laboratories, Albuquerque

    Google Scholar 

  13. Helton JC, Cooke RM, McKay MD, Saltelli A (eds) (2006) Special issue: The fourth international conference on sensitivity analysis of model output – SAMO 2004.Reliab Eng Syst Saf 91:(10–11):1105–1474

    Google Scholar 

  14. Hirota K (1992) An introduction to fuzzy logic applications in intelligent systems. In: Kluwer International Series in Engineering and Computer Science, Chapter probabilistic sets: probabilistic extensions of fuzzy sets, vol 165. Kluwer, Boston, pp 335–354

    Google Scholar 

  15. Klement EP, Puri ML, Ralescu DA (1984) Cybernetics and Systems Research 2, Chapter law of large numbers and central limit theorems for fuzzy random variables.Elsevier, North-Holland, pp 525–529

    Google Scholar 

  16. Kutterer H (2004) Statistical hypothesis tests in case of imprecise data, V Hotine-Marussi Symposium on Mathematical Geodesy.Springer, Berlin, pp 49–56

    Book  Google Scholar 

  17. Li S, Ogura Y (2003) A convergence theorem of fuzzy-valued martingales in the extended hausdorff metric h(inf).Fuzzy Sets Syst 135:391–399

    Article  MathSciNet  MATH  Google Scholar 

  18. Li S, Ogura Y, Nguyen HT (2001) Gaussian processes and martingales for fuzzy valued random variables with continuous parameter.Inf Sci 133:7–21

    Article  MathSciNet  MATH  Google Scholar 

  19. Li S, Ogura Y, Proske FN, Puri ML (2003) Central limit theorems for generalized set-valued random variables.J Math Anal Appl 285:250–263

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu B (2002) Theory and practice of uncertainty programming.Physica, Heidelberg

    Book  Google Scholar 

  21. Liu Y, Qiao Z, Wang G (1997) Fuzzy random reliability of structures based on fuzzy random variables.Fuzzy Sets Syst 86:345–355

    Article  MathSciNet  MATH  Google Scholar 

  22. Möller B, Graf W, Beer M (2003) Safety assessment of structures in view of fuzzy randomness.Comput Struct 81:1567–1582

    Google Scholar 

  23. Möller B, Liebscher M, Schweizerhof K, Mattern S, Blankenhorn G (2008) Structural collapse simulation under consideration of uncertainty – improvement of numerical efficiency. Comput Struct 86(19–20):1875–1884

    Google Scholar 

  24. Montenegro M, Casals MR, Lubiano MA, Gil MA (2001) Two-sample hypothesis tests of means of a fuzzy random variable.Inf Sci 133:89–100

    Article  MathSciNet  MATH  Google Scholar 

  25. Montenegro M, Colubi A, Casals MR, Gil MA (2004) Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable.Metrika 59:31–49

    Article  MathSciNet  MATH  Google Scholar 

  26. Montenegro M, González-Rodríguez G, Gil MA, Colubi A, Casals MR (2004) Soft methodology and random information systems, chapter introduction to ANOVA with fuzzy random variables.Springer, Berlin, pp 487–494

    Google Scholar 

  27. Muhanna RL, Mullen RL (eds) (2004) Proceedings of the NSF workshop on reliable engineering computing. Center for Reliable Engineering Computing.Georgia Tech Savannah, Georgia

    Google Scholar 

  28. Muhanna RL, Mullen RL (eds) (2006) NSF workshop on reliable engineering computing. center for reliable engineering computing. Georgia Tech Savannah, Georgia

    Google Scholar 

  29. Negoita VN, Ralescu DA (1987) Simulation, knowledge-based computing and fuzzy-statistics.Van Nostrand, Reinhold, New York

    MATH  Google Scholar 

  30. Oberguggenberger M, Schuëller GI, Marti K (eds) (2004) Special issue on application of fuzzy sets and fuzzy logic to engineering problems.ZAMM: Z Angew Math Mech 84(10–11):661–776

    Google Scholar 

  31. Okuda T, Tanaka H, Asai K (1978) A formulation of fuzzy decision problems with fuzzy information using probability measures of fuzzy events.Inf Control 38:135–147

    Article  MathSciNet  MATH  Google Scholar 

  32. Proske FN, Puri ML (2002) Strong law of large numbers for banach space valued fuzzy random variables.J Theor Probab 15:543–551

    Article  MathSciNet  MATH  Google Scholar 

  33. Reddy RK, Haldar A (1992) Analysis and management of uncertainty: theory and applications, chapter a random-fuzzy reliability analysis of engineering systems.North-Holland, Amsterdam, pp 319–329

    Google Scholar 

  34. Reddy RK, Haldar A (1992) A random-fuzzy analysis of existing structures. Fuzzy Sets Syst 48:201–210

    Article  Google Scholar 

  35. Ross TJ (2004) Fuzzy logic with engineering applications, 2nd edn. Wiley, Philadelphia

    MATH  Google Scholar 

  36. Ross TJ, Booker JM, Parkinson WJ (eds) (2002) Fuzzy logic and probability applications – bridging the gap.SIAM & ASA, Philadelphia

    MATH  Google Scholar 

  37. Stojakovic M (1994) Fuzzy random variables, expectation, and martingales. J Math Anal Appl 184:594–606

    Article  MathSciNet  MATH  Google Scholar 

  38. Terán P (2004) Cones and decomposition of sub- and supermartingales. Fuzzy Sets Syst 147:465–474

    Google Scholar 

  39. Tonon F, Bernardini A (1998) A random set approach to the optimization of uncertain structures.Comput Struct 68(6):583–600

    Article  MathSciNet  MATH  Google Scholar 

  40. Weichselberger K (2000) The theory of interval-probability as a unifying concept for uncertainty.Int J Approx Reason 24(2–3):149–170

    Article  MathSciNet  MATH  Google Scholar 

  41. Yukari Y, Masao M (1999) Interval and paired probabilities for treating uncertain events.IEICE Trans Inf Syst E82–D(5):955–961

    Google Scholar 

  42. Zadeh L (1985) Is probability theory sufficient for dealing with uncertainty in ai: A negative view.In: Proceedings of the 1st Annual Conference on Uncertainty in Artificial Intelligence (UAI-85).Elsevier Science, New York, pp 103–116

    Google Scholar 

  43. Zhang Y, Wang G, Su F (1996) The general theory for response analysis of fuzzy stochastic dynamical systems.Fuzzy Sets Syst 83:369–405

    Article  MathSciNet  MATH  Google Scholar 

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  1. National University of Singapore, Kent Ridge, Singapore

    Michael Beer

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Beer, M. (2012). Fuzzy Probability Theory. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_76

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