Fuzzy Probability and Statistics (eBook)

Contents 7
Chapter 1 Introduction 14
1.1 Introduction 14
1.2 Notation 16
1.3 Previous Research 17
1.4 Figures 17
1.5 Maple/Solver Commands 18
1.6 References 18
Chapter 2 Fuzzy Sets 20
2.1 Introduction 20
2.2 Fuzzy Sets 20
2.3 Fuzzy Arithmetic 24
2.4 Fuzzy Functions 26
2.5 Ordering Fuzzy Numbers 30
2.6 References 31
Chapter 3 Fuzzy Probability Theory 33
3.1 Introduction 33
3.2 Fuzzy Probabilities from Con . dence Intervals 33
3.3 Fuzzy Probabilities from Expert Opinion 35
3.4 Restricted Fuzzy Arithmetic 36
3.5 Fuzzy Probability 44
3.6 Fuzzy Conditional Probability 48
3.7 Fuzzy Independence 50
3.8 Fuzzy Bayes’ Formula 52
3.9 Applications 53
3.10 References 60
Chapter 4 Discrete Fuzzy Random Variables 62
4.1 Introduction 62
4.2 Fuzzy Binomial 62
4.3 Fuzzy Poisson 65
4.4 Applications 68
4.5 References 71
Chapter 5 Continuous Fuzzy Random Variables 72
5.1 Introduction 72
5.2 Fuzzy Uniform 72
5.3 Fuzzy Normal 74
5.4 Fuzzy Negative Exponential 76
5.5 Applications 78
5.6 References 85
Chapter 6 Estimate µ, Variance Known 86
6.1 Introduction 86
6.2 Fuzzy Estimation 86
6.3 Fuzzy Estimator of 87
6.4 References 90
Chapter 7 Estimate µ, Variance Unknown 91
7.1 Fuzzy Estimator of 91
7.2 References 93
Chapter 8 Estimate p, Binomial Population 94
8.1 Fuzzy Estimator of 94
8.2 References 96
Chapter 9 Estimate s2 from a Normal Population 97
9.1 Introduction 97
9.2 Biased Fuzzy Estimator 97
9.3 Unbiased Fuzzy Estimator 98
9.4 References 102
Chapter 10 Fuzzy Arrival/Service Rates 103
10.1 Introduction 103
10.2 Fuzzy Arrival Rate 103
10.3 Fuzzy Service Rate 105
10.4 References 107
Chapter 11 Fuzzy Uniform 108
11.1 Introduction 108
11.2 Fuzzy Estimators 108
11.3 References 112
Chapter 12 Fuzzy Max Entropy Principle 113
12.1 Introduction 113
12.2 Maximum Entropy Principle 113
12.3 Imprecise Side-Conditions 117
12.4 Summary and Conclusions 119
12.5 References 120
Chapter 13 Max Entropy: Crisp Discrete Solutions 121
13.1 Introduction 121
13.2 Max Entropy: Discrete Distributions 121
13.3 Max Entropy: Imprecise Side-Conditions 122
13.4 Summary and Conclusions 129
13.5 References 129
Chapter 14 Max Entropy: Crisp Continuous Solutions 131
14.1 Introduction 131
14.2 Max Entropy: Probability Densities 132
14.3 Max Entropy: Imprecise Side-Conditions 133
14.4 E = [0,M] 133
14.5 E = [0,8) 141
14.6 E = (.8,8) 145
14.7 Summary and Conclusions 146
14.8 References 146
Chapter 15 Tests on µ, Variance Known 148
15.1 Introduction 148
15.2 Non-Fuzzy Case 148
15.3 Fuzzy Case 149
15.4 One-Sided Tests 153
15.5 References 154
Chapter 16 Tests on µ, Variance Unknown 155
16.1 Introduction 155
16.2 Crisp Case 155
16.3 Fuzzy Model 156
16.4 References 161
Chapter 17 Tests on p for a Binomial Population 162
17.1 Introduction 162
17.2 Non-Fuzzy Test 162
17.3 Fuzzy Test 163
17.4 References 165
Chapter 18 Tests on s2, Normal Population 166
18.1 Introduction 166
18.2 Crisp Hypothesis Test 166
18.3 Fuzzy Hypothesis Test 167
18.4 References 169
Chapter 19 Fuzzy Correlation 170
19.1 Introduction 170
19.2 Crisp Results 170
19.3 Fuzzy Theory 171
19.4 References 173
Chapter 20 Estimation in Simple Linear Regression 174
20.1 Introduction 174
20.2 Fuzzy Estimators 175
20.3 References 178
Chapter 21 Fuzzy Prediction in Linear Regression 179
21.1 Prediction 179
21.2 References 181
Chapter 22 Hypothesis Testing in Regression 182
22.1 Introduction 182
22.2 Tests on 182
22.3 Tests on 184
22.4 References 186
Chapter 23 Estimation in Multiple Regression 187
23.1 Introduction 187
23.2 Fuzzy 188
23.3 References 192
Chapter 24 Fuzzy Prediction in Regression 193
24.1 Prediction 193
24.2 References 195
Chapter 25 Hypothesis Testing in Regression 196
25.1 Introduction 196
25.2 Tests on 196
25.3 Tests on 198
25.4 References 200
Chapter 26 Fuzzy One-Way ANOVA 201
26.1 Introduction 201
26.2 Crisp Hypothesis Test 201
26.3 Fuzzy Hypothesis Test 202
26.4 References 205
Chapter 27 Fuzzy Two-Way ANOVA 206
27.1 Introduction 206
27.2 Crisp Hypothesis Tests 206
27.3 Fuzzy Hypothesis Tests 208
27.4 References 214
Chapter 28 Fuzzy Estimator for the Median 215
28.1 Introduction 215
28.2 Crisp Estimator for the Median 215
28.3 Fuzzy Estimator 216
28.4 Reference 217
Chapter 29 Random Fuzzy Numbers 218
29.1 Introduction 218
29.2 Random Fuzzy Numbers 219
29.3 Tests for Randomness 220
29.4 Monte Carlo Study 225
29.5 References 228
Chapter 30 Selected Maple/Solver Commands 230
30.1 Introduction 230
30.2 SOLVER 230
30.3 Maple 232
30.4 References 246
Chapter 31 Summary and Future Research 247
31.1 Summary 247
31.2 Future Research 248
31.3 References 250
Index 251
List of Figures 258
List of Tables 261

Chapter 1
Introduction (p. 1-2)

1.1 Introduction

This book is written in the following divisions: (1) the introductory chapters consisting of Chapters 1 and 2, (2) introduction to fuzzy probability in Chapters 3-5, (3) introduction to fuzzy estimation in Chapters 6-11, (4) fuzzy/crisp estimators of probability density (mass) functions based on a fuzzy maximum entropy principle in Chapters 12-14, (5) introduction to fuzzy hypothesis testing in Chapters 15-18, (6) fuzzy correlation and regression in Chapters 19-25, (7) Chapters 26 and 27 are about a fuzzy ANOVA model, (8) a fuzzy estimator of the median in nonparametric statistics in Chapter 28, and (9) random fuzzy numbers with applications to Monte Carlo studies in Chapter 29. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. For a beginning introduction to fuzzy sets and fuzzy logic see [8]. One other item relating to fuzzy sets, needed in fuzzy hypothesis testing, is also in Chapter 2: how we will determine which of the following three possibilities is true M <, N, M >, N or M . N, for two fuzzy numbers M, N.

The introduction to fuzzy probability in Chapters 3-5 is based on the book [1] and the reader is referred to that book for more information, especially applications. What is new here is: (1) using a nonlinear optimization program in Maple [13] to solve certain optimization problems in fuzzy probability, where previously we used a graphical method, and (2) a new algorithm, suitable for using only pencil and paper, for solving some restricted fuzzy arithmetic problems.

The introduction to fuzzy estimation is based on the book [3] and we refer the interested reader to that book for more about fuzzy estimators. The fuzzy estimators omitted from this book are those for µ1 . µ2, p1 . p2, s1/s2, etc. Fuzzy estimators for arrival and service rates is from [2] and [4]. The reader should see those book for applications in queuing networks. Also, fuzzy estimators for the uniform probability density can be found in [4], but the derivation of these fuzzy estimators is new to this book. The fuzzy uniform distribution was used for arrival/service rates in queuing models in [4].

The fuzzy/crisp probability density estimators based on a fuzzy maximum entropy principle are based on the papers [5],[6] and [7] and are new to this book. In Chapter 12 we obtain fuzzy results but in Chapters 13 and 14 we determine crisp discrete and crisp continuous probability densities. The introduction to fuzzy hypothesis testing in Chapters 15-18 is based on the book [3] and the reader needs to consult that book for more fuzzy hypothesis testing. What we omitted are tests on µ1 = µ2, p1 = p2, s1 = s2, etc.

The chapters on fuzzy correlation and regression come from [3]. The results on the fuzzy ANOVA (Chapters 26 and 27) and a fuzzy estimator for the median (Chapter 28) are new and have not been published before. The chapter on random fuzzy numbers (Chapter 29) is also new to this book and these results have not been previously published. Applications of crisp random numbers to Monte Carlo studies are well known and we also plan to use random fuzzy numbers in Monte Carlo studies. Our first use of random fuzzy numbers will be to get approximate solutions to fuzzy optimization problems whose solution is unknown or computationally very difficult. However, this becomes a rather large project and will probably be the topic of a future book.

Chapter 30 contains selected Maple/Solver ([11],[13],[20]) commands used in the book to solve optimization problems or to generate the figures. The final chapter has a summary and suggestions for future research. All chapters can be read independently. This means that some material is repeated in a sequence of chapters. For example, in Chapters 15-18 on fuzzy hypothesis testing in each chapter we first review the crisp case, then fuzzify to obtain our fuzzy statistic which is then used to construct the fuzzy critical values and we finally present a numerical example. However, you should first know about fuzzy estimators (Chapters 6-11) before going on to fuzzy hypothesis testing.

A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data. Other authors discussing fuzzy statistics usually begin with fuzzy data. We assume we have a random sample giving real number data x1, x2, ..., xn which is then used to generate our fuzzy estimators. Using fuzzy estimators in hypothesis testing and regression obviously leads to fuzzy hypothesis testing and fuzzy regression.