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 before Includes supplementary material: sn.pub/extras

Inhalt
1 Introduction.- 1.1 Introduction.- 1.2 References.- 2 Fuzzy Sets.- 2.1 Introduction.- 2.2 Fuzzy Sets.- 2.3 Fuzzy Arithmetic.- 2.4 Fuzzy Functions.- 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.- 2.10 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.- 3.7 References.- 4 Discrete Fuzzy Random Variables.- 4.1 Introduction.- 4.2 Fuzzy Binomial.- 4.3 Fuzzy Poisson.- 4.4 Applications.- 4.5 References.- 5 Fuzzy Queuing Theory.- 5.1 Introduction.- 5.2 Regular, Finite, Markov Chains.- 5.3 Fuzzy Queuing Theory.- 5.4 Applications.- 5.5 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.- 8.6 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.- 10.3 References.- 11 Applications of Joint Distributions.- 11.1 Introduction.- 11.2 Political Polls.- 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.- 13.1Introduction.- 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.- 16.3 Research Agenda.- 16.4 Conclusions.- List of Figures.- List of Tables.