Fuzzy Probabilities

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.

New method of dealing with imprecise probabilities, most of which not published beforeIncludes supplementary material: sn.pub/extras

Inhaltsangabe1 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.