Fuzzy Probabilities: New Approach and Applications

ISBN-13: 9783642867880 / Angielski / Miękka / 2012 / 165 str.

James J. Buckley; James J. Buckley

Opis

In probability and statistics we often have to estimate probabilities and parameters in probability distributions using a random sample. Instead of using a point estimate calculated from the data we propose using fuzzy numbers which are constructed from a set of confidence intervals. In probability calculations we apply constrained fuzzy arithmetic because probabilities must add to one. Fuzzy random variables have fuzzy distributions. A fuzzy normal random variable has the normal distribution with fuzzy number mean and variance. Applications are to queuing theory, Markov chains, inventory control, decision theory and reliability theory.

Spis treści

1 Introduction.- 1.1 Introduction.- 1.2 References.- 2 Fuzzy Sets.- 2.1 Introduction.- 2.2 Fuzzy Sets.- 2.2.1 Fuzzy Numbers.- 2.2.2 Alpha-Cuts.- 2.2.3 Inequalities.- 2.2.4 Discrete Fuzzy Sets.- 2.3 Fuzzy Arithmetic.- 2.3.1 Extension Principle.- 2.3.2 Interval Arithmetic.- 2.3.3 Fuzzy Arithmetic.- 2.4 Fuzzy Functions.- 2.4.1 Extension Principle.- 2.4.2 Alpha-Cuts and Interval Arithmetic.- 2.4.3 Differences.- 2.5 Finding the Minimum of a Fuzzy Number.- 2.6 Ordering Fuzzy Numbers.- 2.7 Fuzzy Probabilities.- 2.8 Fuzzy Numbers from Confidence Intervals.- 2.9 Computing Fuzzy Probabilities.- 2.9.1 First Problem.- 2.9.2 Second Problem.- 2.10 Figures.- 2.11 References.- 3 Fuzzy Probability Theory.- 3.1 Introduction.- 3.2 Fuzzy Probability.- 3.3 Fuzzy Conditional Probability.- 3.4 Fuzzy Independence.- 3.5 Fuzzy Bayes’ Formula.- 3.6 Applications.- 3.6.1 Blood Types.- 3.6.2 Resistance to Surveys.- 3.6.3 Testing for HIV.- 3.6.4 Color Blindness.- 3.6.5 Fuzzy Bayes.- 3.7 References.- 4 Discrete Fuzzy Random Variables.- 4.1 Introduction.- 4.2 Fuzzy Binomial.- 4.3 Fuzzy Poisson.- 4.4 Applications.- 4.4.1 Fuzzy Poisson Approximating Fuzzy Binomial.- 4.4.2 Overbooking.- 4.4.3 Rapid Response Team.- 4.5 References.- 5 Fuzzy Queuing Theory.- 5.1 Introduction.- 5.2 Regular, Finite, Markov Chains.- 5.3 Fuzzy Queuing Theory.- 5.4 Applications.- 5.4.1 Machine Servicing Problem.- 5.4.2 Fuzzy Queuing Decision Problem.- 5.5 References.- 6 Fuzzy Markov Chains.- 6.1 Introduction.- 6.2 Regular Markov Chains.- 6.3 Absorbing Markov Chains.- 6.4 Application: Decision Model.- 6.5 References.- 7 Fuzzy Decisions Under Risk.- 7.1 Introduction.- 7.2 Without Data.- 7.3 With Data.- 7.4 References.- 8 Continuous Fuzzy Random Variables.- 8.1 Introduction.- 8.2 Fuzzy Uniform.- 8.3 Fuzzy Normal.- 8.4 Fuzzy Negative Exponential.- 8.5 Applications.- 8.5.1 Fuzzy Uniform.- 8.5.2 Fuzzy Normal Approximation to Fuzzy Binomial.- 8.5.3 Fuzzy Normal Approximation to Fuzzy Poisson.- 8.5.4 Fuzzy Normal.- 8.5.5 Fuzzy Negative Exponential.- 8.6 References.- 9 Fuzzy Inventory Control.- 9.1 Introduction.- 9.2 Single Period Model.- 9.3 Multiple Periods.- 9.4 References.- 10 Joint Fuzzy Probability Distributions.- 10.1 Introduction.- 10.2 Continuous Case.- 10.2.1 Fuzzy Marginals.- 10.2.2 Fuzzy Conditionals.- 10.2.3 Fuzzy Correlation.- 10.2.4 Fuzzy Bivariate Normal.- 10.3 References.- 11 Applications of Joint Distributions.- 11.1 Introduction.- 11.2 Political Polls.- 11.2.1 Fuzzy Marginals.- 11.2.2 Fuzzy Conditionals.- 11.2.3 Fuzzy Correlation.- 11.3 Fuzzy Reliability Theory.- 11.4 References.- 12 Functions of a Fuzzy Random Variable.- 12.1 Introduction.- 12.2 Discrete Fuzzy Random Variables.- 12.3 Continuous Fuzzy Random Variables.- 13 Functions of Fuzzy Random Variables.- 13.1 Introduction.- 13.2 One-to-One Transformation.- 13.3 Other Transformations.- 14 Law of Large Numbers.- 15 Sums of Fuzzy Random Variables.- 15.1 Introduction.- 15.2 Sums.- 16 Conclusions and Future Research.- 16.1 Introduction.- 16.2 Summary.- 16.2.1 Chapter 3.- 16.2.2 Chapter 4.- 16.2.3 Chapter 5.- 16.2.4 Chapter 6.- 16.2.5 Chapter 7.- 16.2.6 Chapter 8.- 16.2.7 Chapter 9.- 16.2.8 Chapter 10.- 16.2.9 Chapter 11.- 16.2.10 Chapter 12.- 16.2.11 Chapter 13.- 16.2.12 Chapter 14.- 16.2.13 Chapter 15.- 16.3 Research Agenda.- 16.3.1 Chapter 3.- 16.3.2 Chapter 4.- 16.3.3 Chapter 5.- 16.3.4 Chapter 6.- 16.3.5 Chapter 7.- 16.3.6 Chapter 8.- 16.3.7 Chapter 9.- 16.3.8 Chapter 10.- 16.3.9 Chapter 11.- 16.3.10 Chapter 12.- 16.3.11 Chapter 13.- 16.3.12 Chapter 14.- 16.3.13 Chapter 15.- 16.4 Conclusions.- List of Figures.- List of Tables.