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小木虫论坛-学术科研互动平台 » 专业学科区 » 数学 » 概率论与统计 » Statistical Distributions 4ed_Catherine Forbes
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[资源] Statistical Distributions 4ed_Catherine Forbes

Contents
Preface xvii
1. Introduction 1
2. Terms and Symbols 3
2.1 Probability, Random Variable, Variate, and Number 3
Probabilistic Experiment 3
Sample Space 3
Random Variable 3
Variate 3
Random Number 4
2.2 Range, Quantile, Probability Statement, and Domain 4
Range 4
Quantile 5
Probability Statement 5
Probability Domain 5
2.3 Distribution Function and Survival Function 5
Distribution Function 5
Survival Function 6
2.4 Inverse Distribution Function and Inverse Survival Function 7
Inverse Survival Function 8
2.5 Probability Density Function and Probability Function 8
2.6 Other Associated Functions and Quantities 9
3. General Variate Relationships 15
3.1 Introduction 15
3.2 Function of a Variate 15
3.3 One-to-One Transformations and Inverses 16
Inverse of a One-to-One Function 17
3.4 Variate Relationships Under One-to-One Transformation 17
Probability Statements 17
Distribution Function 17
Inverse Distribution Function 18
Equivalence of Variates 18
Inverse Function of a Variate 18
vii
viii Contents
3.5 Parameters, Variate, and Function Notation 19
Variate and Function Notation 19
3.6 Transformation of Location and Scale 20
3.7 Transformation from the Rectangular Variate 20
3.8 Many-to-One Transformations 22
Symmetrical Distributions 22
4. Multivariate Distributions 24
4.1 Joint Distributions 24
Joint Range 24
Bivariate Quantile 24
Joint Probability Statement 24
Joint Probability Domain 25
Joint Distribution Function 25
Joint Probability Density Function 25
Joint Probability Function 25
4.2 Marginal Distributions 26
Marginal Probability Density Function and Marginal
Probability Function 26
4.3 Independence 27
4.4 Conditional Distributions 28
Conditional Probability Function and Conditional
Probability Density Function 28
Composition 29
4.5 Bayes’ Theorem 30
4.6 Functions of a Multivariate 30
5. Stochastic Modeling 32
5.1 Introduction 32
5.2 Independent Variates 32
5.3 Mixture Distributions 33
Finite Mixture 33
Infinite Mixture of Distributions 35
5.4 Skew-Symmetric Distributions 38
5.5 Distributions Characterized by Conditional Skewness 39
5.6 Dependent Variates 42
6. Parameter Inference 44
6.1 Introduction 44
6.2 Method of Percentiles Estimation 44
6.3 Method of Moments Estimation 45
6.4 Maximum Likelihood Inference 47
Properties of MLEs 47
Approximate Sampling Distribution for Fixed n 48
Contents ix
6.5 Bayesian Inference 50
Marginal Posteriors 51
7. Bernoulli Distribution 53
7.1 Random Number Generation 53
7.2 Curtailed Bernoulli Trial Sequences 53
7.3 Urn Sampling Scheme 54
7.4 Note 54
8. Beta Distribution 55
8.1 Notes on Beta and Gamma Functions 56
Definitions 56
Interrelationships 56
Special Values 57
Alternative Expressions 57
8.2 Variate Relationships 57
8.3 Parameter Estimation 59
8.4 Random Number Generation 60
8.5 Inverted Beta Distribution 60
8.6 Noncentral Beta Distribution 61
8.7 Beta Binomial Distribution 61
9. Binomial Distribution 62
9.1 Variate Relationships 64
9.2 Parameter Estimation 65
9.3 Random Number Generation 65
10. Cauchy Distribution 66
10.1 Note 66
10.2 Variate Relationships 67
10.3 Random Number Generation 68
10.4 Generalized Form 68
11. Chi-Squared Distribution 69
11.1 Variate Relationships 71
11.2 Random Number Generation 72
11.3 Chi Distribution 73
12. Chi-Squared (Noncentral) Distribution 74
12.1 Variate Relationships 75
x Contents
13. Dirichlet Distribution 77
13.1 Variate Relationships 77
13.2 Dirichlet Multinomial Distribution 78
14. Empirical Distribution Function 79
14.1 Estimation from Uncensored Data 79
14.2 Estimation from Censored Data 79
14.3 Parameter Estimation 81
14.4 Example 81
14.5 Graphical Method for the Modified Order-Numbers 81
14.6 Model Accuracy 83
15. Erlang Distribution 84
15.1 Variate Relationships 85
15.2 Parameter Estimation 85
15.3 Random Number Generation 85
16. Error Distribution 86
16.1 Note 87
16.2 Variate Relationships 87
17. Exponential Distribution 88
17.1 Note 89
17.2 Variate Relationships 91
17.3 Parameter Estimation 92
17.4 Random Number Generation 92
18. Exponential Family 93
18.1 Members of the Exponential Family 93
18.2 Univariate One-Parameter Exponential Family 93
18.3 Parameter Estimation 95
18.4 Generalized Exponential Distributions 95
Generalized Student’s t Distribution 95
Variate Relationships 96
Generalized Exponential Normal Distribution 96
Generalized Lognormal Distribution 96
Variate Relationships 97
Contents xi
19. Extreme Value (Gumbel) Distribution 98
19.1 Note 99
19.2 Variate Relationships 100
19.3 Parameter Estimation 101
19.4 Random Number Generation 101
20. F (Variance Ratio) or Fisher–Snedecor Distribution 102
20.1 Variate Relationships 103
21. F (Noncentral) Distribution 107
21.1 Variate Relationships 108
22. Gamma Distribution 109
22.1 Variate Relationships 110
22.2 Parameter Estimation 111
22.3 Random Number Generation 112
22.4 Inverted Gamma Distribution 112
22.5 Normal Gamma Distribution 112
22.6 Generalized Gamma Distribution 113
Variate Relationships 113
23. Geometric Distribution 114
23.1 Notes 115
23.2 Variate Relationships 115
23.3 Random Number Generation 116
24. Hypergeometric Distribution 117
24.1 Note 118
24.2 Variate Relationships 118
24.3 Parameter Estimation 118
24.4 Random Number Generation 119
24.5 Negative Hypergeometric Distribution 119
24.6 Generalized Hypergeometric Distribution 119
25. Inverse Gaussian (Wald) Distribution 120
25.1 Variate Relationships 121
25.2 Parameter Estimation 121
xii Contents
26. Laplace Distribution 122
26.1 Variate Relationships 124
26.2 Parameter Estimation 124
26.3 Random Number Generation 124
27. Logarithmic Series Distribution 125
27.1 Variate Relationships 126
27.2 Parameter Estimation 126
28. Logistic Distribution 127
28.1 Notes 128
28.2 Variate Relationships 128
28.3 Parameter Estimation 130
28.4 Random Number Generation 130
29. Lognormal Distribution 131
29.1 Variate Relationships 132
29.2 Parameter Estimation 134
29.3 Random Number Generation 134
30. Multinomial Distribution 135
30.1 Variate Relationships 136
30.2 Parameter Estimation 136
31. Multivariate Normal (Multinormal) Distribution 137
31.1 Variate Relationships 138
31.2 Parameter Estimation 138
32. Negative Binomial Distribution 139
32.1 Note 140
32.2 Variate Relationships 141
32.3 Parameter Estimation 142
32.4 Random Number Generation 142
33. Normal (Gaussian) Distribution 143
33.1 Variate Relationships 144
33.2 Parameter Estimation 147
Contents xiii
33.3 Random Number Generation 147
33.4 Truncated Normal Distribution 147
33.5 Variate Relationships 148
34. Pareto Distribution 149
34.1 Note 149
34.2 Variate Relationships 150
34.3 Parameter Estimation 151
34.4 Random Number Generation 151
35. Poisson Distribution 152
35.1 Note 153
35.2 Variate Relationships 153
35.3 Parameter Estimation 156
35.4 Random Number Generation 156
36. Power Function Distribution 157
36.1 Variate Relationships 157
36.2 Parameter Estimation 159
36.3 Random Number Generation 159
37. Power Series (Discrete) Distribution 160
37.1 Note 160
37.2 Variate Relationships 161
37.3 Parameter Estimation 161
38. Queuing Formulas 162
38.1 Characteristics of Queuing Systems and Kendall-Lee Notation 162
Characteristics of Queuing Systems 162
Kendall-Lee Notation 164
38.2 Definitions, Notation, and Terminology 164
Steady State 164
Traffic Intensity and Traffic Density 164
Notation and Terminology 164
38.3 General Formulas 166
38.4 Some Standard Queuing Systems 166
The M/M/1/G/∞/∞ System 166
The M/M/s/G/ ∞/∞ System 166
The M/G/1/G/∞/∞ System (Pollaczek-Khinchin) 168
The M/M/1/G/m/∞ System 168
The M/G/m/G/m/∞ System (Erlang) 169
xiv Contents
The M/M/1/G/N/N System (One Server, Finite Population N) 169
The M/M/s/G/N/N System (s Servers, Finite Population N) 171
39. Rayleigh Distribution 173
39.1 Variate Relationships 173
39.2 Parameter Estimation 175
40. Rectangular (Uniform) Continuous Distribution 176
40.1 Variate Relationships 177
40.2 Parameter Estimation 179
40.3 Random Number Generation 179
41. Rectangular (Uniform) Discrete Distribution 180
41.1 General Form 181
41.2 Parameter Estimation 182
42. Student’s t Distribution 183
42.1 Variate Relationships 185
42.2 Random Number Generation 186
43. Student’s t (Noncentral) Distribution 187
43.1 Variate Relationships 188
44. Triangular Distribution 189
44.1 Variate Relationships 189
44.2 Random Number Generation 190
45. von Mises Distribution 191
45.1 Note 191
45.2 Variate Relationships 192
45.3 Parameter Estimation 192
46. Weibull Distribution 193
46.1 Note 195
46.2 Variate Relationships 196
46.3 Parameter Estimation 196
46.4 Random Number Generation 196
Contents xv
46.5 Three-Parameter Weibull Distribution 196
46.6 Three-Parameter Weibull Random Number Generation 198
46.7 Bi-Weibull Distribution 198
46.8 Five-Parameter Bi-Weibull Distribution 198
Bi-Weibull Random Number Generation 200
Bi-Weibull Graphs 200
46.9 Weibull Family 201
47. Wishart (Central) Distribution 202
47.1 Note 203
47.2 Variate Relationships 203
48. Statistical Tables 204
Table48.1: Normal Distribution Function −F N (x) 205
Table48.2: Percentiles of the Chi-Squared χ 2 : ν Distribution, G(1 − α) 206
Table48.3: Percentiles of the F : ν,ω Distribution 207
Table48.4: Percentiles of the Student’s t Distribution 209
Table48.5: Partial Expectations for the Standard Normal Distribution 210
Bibliography 211
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