Download Statistical Distributions 4th Edition PDF

Statistical Distributions 4th Edition pdf

Statistical Distributions 4th Edition – Fully updated to reflect the latest developments on the topic, Statistical Distributions, Fourth Edition continues to serve as an authoritative guide on the application of statistical methods to research across various disciplines. The book provides a concise presentation of popular statistical distributions along with the necessary knowledge for their successful use in data modeling and analysis.

Following a basic introduction, forty popular distributions are outlined in individual chapters that are complete with related facts and formulas. Reflecting the latest changes and trends in statistical distribution theory, the Fourth Edition features:

  • A new chapter on queuing formulas that discusses standard formulas that often arise from simple queuing systems
  • Methods for extending independent modeling schemes to the dependent case, covering techniques for generating complex distributions from simple distributions
  • New coverage of conditional probability, including conditional expectations and joint and marginal distributions
  • Commonly used tables associated with the normal (Gaussian), student-t, F and chi-square distributions
  • Additional reviewing methods for the estimation of unknown parameters, such as the method of percentiles, the method of moments, maximum likelihood inference, and Bayesian inference

Statistical Distributions, Fourth Edition is an excellent supplement for upper-undergraduate and graduate level courses on the topic. It is also a valuable reference for researchers and practitioners in the fields of engineering, economics, operations research, and the social sciences who conduct statistical analyses.

Content – Statistical Distributions 4th Edition

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

Statistical Distributions 4th Edition

  • Probability Statements 17
  • Distribution Function 17
  • Inverse Distribution Function 18
  • Equivalence of Variates 18
  • Inverse Function of a Variate 18

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

  1. 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
  2. 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
  3. 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

6.5 Bayesian Inference 50
Marginal Posteriors 51 Statistical Distributions 4th Edition

  1. 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
  2. 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
  3. Binomial Distribution 62
    9.1 Variate Relationships 64
    9.2 Parameter Estimation 65
    9.3 Random Number Generation 65
  4. Cauchy Distribution 66
    10.1 Note 66
    10.2 Variate Relationships 67
    10.3 Random Number Generation 68
    10.4 Generalized Form 68
  5. Chi-Squared Distribution 69
    11.1 Variate Relationships 71
    11.2 Random Number Generation 72
    11.3 Chi Distribution 73
  6. Chi-Squared (Noncentral) Distribution 74
    12.1 Variate Relationships 75
    Statistical Distributions 4th Edition
  1. Dirichlet Distribution 77
  2. 13.1 Variate Relationships 77
  3. 13.2 Dirichlet Multinomial Distribution 78
  4. Empirical Distribution Function 79
  5. 14.1 Estimation from Uncensored Data 79
  6. 14.2 Estimation from Censored Data 79
  7. 14.3 Parameter Estimation 81
  8. 14.4 Example 81
  9. 14.5 Graphical Method for the Modified Order-Numbers 81
  10. 14.6 Model Accuracy 83
  11. Erlang Distribution 84
  12. 15.1 Variate Relationships 85
  13. 15.2 Parameter Estimation 85
  14. 15.3 Random Number Generation 85
  15. Error Distribution 86
  16. 16.1 Note 87
  17. 16.2 Variate Relationships 87
  18. Exponential Distribution 88
  19. 17.1 Note 89
  20. 17.2 Variate Relationships 91
  21. 17.3 Parameter Estimation 92
  22. 17.4 Random Number Generation 92
  23. Exponential Family 93
  24. 18.1 Members of the Exponential Family 93
  25. 18.2 Univariate One-Parameter Exponential Family 93
  26. 18.3 Parameter Estimation 95
  27. 18.4 Generalized Exponential Distributions 95
  28. Generalized Student’s t Distribution 95
  29. Variate Relationships 96
  30. Generalized Exponential Normal Distribution 96
  31. Generalized Lognormal Distribution 96
  32. Variate Relationships 97 Extreme Value (Gumbel) Distribution 98
  33. 19.1 Note 99
  34. 19.2 Variate Relationships 100
  35. 19.3 Parameter Estimation 101
  36. 19.4 Random Number Generation 101
  37. F (Variance Ratio) or Fisher–Snedecor Distribution 102
  38. 20.1 Variate Relationships 103
  39. F (Noncentral) Distribution 107
  40. 21.1 Variate Relationships 108
  41. Gamma Distribution 109
  42. 22.1 Variate Relationships 110
  43. 22.2 Parameter Estimation 111
  44. 22.3 Random Number Generation 112
  45. 22.4 Inverted Gamma Distribution 112
  46. 22.5 Normal Gamma Distribution 112
  47. 22.6 Generalized Gamma Distribution 113
  48. Variate Relationships 113
  49. Geometric Distribution 114
  50. 23.1 Notes 115
  51. 23.2 Variate Relationships 115
  52. 23.3 Random Number Generation 116
  53. Hypergeometric Distribution 117
  54. 24.1 Note 118
  55. 24.2 Variate Relationships 118
  56. 24.3 Parameter Estimation 118
  57. 24.4 Random Number Generation 119
  58. 24.5 Negative Hypergeometric Distribution 119
  59. 24.6 Generalized Hypergeometric Distribution 119
  60. Inverse Gaussian (Wald) Distribution 120
  61. 25.1 Variate Relationships 121
  62. 25.2 Parameter Estimation 121 Laplace Distribution 122
  63. 26.1 Variate Relationships 124
  64. 26.2 Parameter Estimation 124
  65. 26.3 Random Number Generation 124
  66. Logarithmic Series Distribution 125
  67. 27.1 Variate Relationships 126
  68. 27.2 Parameter Estimation 126
  69. Logistic Distribution 127
  70. 28.1 Notes 128
  71. 28.2 Variate Relationships 128
  72. 28.3 Parameter Estimation 130
  73. 28.4 Random Number Generation 130
  74. Lognormal Distribution 131
  75. 29.1 Variate Relationships 132
  76. 29.2 Parameter Estimation 134
  77. 29.3 Random Number Generation 134
  78. Multinomial Distribution 135
  79. 30.1 Variate Relationships 136
  80. 30.2 Parameter Estimation 136
  81. Multivariate Normal (Multinormal) Distribution 137
  82. 31.1 Variate Relationships 138
  83. 31.2 Parameter Estimation 138
  84. Negative Binomial Distribution 139
  85. 32.1 Note 140
  86. 32.2 Variate Relationships 141
  87. 32.3 Parameter Estimation 142
  88. 32.4 Random Number Generation 142
  89. Normal (Gaussian) Distribution 143
  90. 33.1 Variate Relationships 144
  91. 33.2 Parameter Estimation 147

33.3 Random Number Generation 147
33.4 Truncated Normal Distribution 147
33.5 Variate Relationships 148 Statistical Distributions 4th Edition

  1. Pareto Distribution 149
    34.1 Note 149
    34.2 Variate Relationships 150
    34.3 Parameter Estimation 151
    34.4 Random Number Generation 151
  2. Poisson Distribution 152
    35.1 Note 153
    35.2 Variate Relationships 153
    35.3 Parameter Estimation 156
    35.4 Random Number Generation 156
  3. Power Function Distribution 157
    36.1 Variate Relationships 157
    36.2 Parameter Estimation 159
    36.3 Random Number Generation 159
  4. Power Series (Discrete) Distribution 160
    37.1 Note 160
    37.2 Variate Relationships 161
    37.3 Parameter Estimation 161
  5. Queuing Formulas 162 Statistical Distributions 4th Edition
    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

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

  1. Rayleigh Distribution 173 Statistical Distributions 4th Edition
    39.1 Variate Relationships 173
    39.2 Parameter Estimation 175
  2. Rectangular (Uniform) Continuous Distribution 176
    40.1 Variate Relationships 177
    40.2 Parameter Estimation 179
    40.3 Random Number Generation 179
  3. Rectangular (Uniform) Discrete Distribution 180
    41.1 General Form 181
    41.2 Parameter Estimation 182
  4. Student’s t Distribution 183
    42.1 Variate Relationships 185
    42.2 Random Number Generation 186
  5. Student’s t (Noncentral) Distribution 187
    43.1 Variate Relationships 188
  6. Triangular Distribution 189
    44.1 Variate Relationships 189
    44.2 Random Number Generation 190
  7. von Mises Distribution 191
    45.1 Note 191
    45.2 Variate Relationships 192
    45.3 Parameter Estimation 192
  8. Weibull Distribution 193
    46.1 Note 195
    46.2 Variate Relationships 196
    46.3 Parameter Estimation 196
    46.4 Random Number Generation 196 Statistical Distributions 4th Edition

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

  1. Wishart (Central) Distribution 202
    47.1 Note 203
    47.2 Variate Relationships 203
  2. Statistical Tables 204
    Table 48.1: Normal Distribution Function −FN(x) 205
    Table 48.2: Percentiles of the Chi-Squared χ2 : ν Distribution, G(1 − α) 206
    Table 48.3: Percentiles of the F : ν, ω Distribution 207
    Table 48.4: Percentiles of the Student’s t Distribution 209
    Table 48.5: Partial Expectations for the Standard Normal Distribution 210

Preface – Statistical Distributions 4th Edition

This revised handbook provides a concise summary of the salient facts and formulas relating to 40 major probability distributions, together with associated diagrams that allow the shape and other general properties of each distribution to be readily appreciated. In the introductory chapters the fundamental concepts of the subject are covered with clarity, and the rules governing the relationships between variates are described. Statistical Distributions 4th Edition

Extensive use is made of the inverse distribution function and a definition establishes a variate as a generalized form of a random variable. A consistent and unambiguous system of nomenclature can thus be developed, with chapter summaries relating to individual distributions. Students, teachers, and practitioners for whom statistics is either a primary or secondary discipline will find this book of great value, both for factual references and as a guide to the basic principles of the subject. It fulfills the need for rapid access to information that must otherwise be gleaned from many scattered sources. Statistical Distributions 4th Edition

The first version of this book, written by N. A. J. Hastings and J. B. Peacock, was published by Butterworths, London, 1975. The second edition, with a new author, M. A. Evans, was published by John Wiley & Sons in 1993, with a third edition by the same authors published by John Wiley & Sons in 2000. This fourth edition sees the addition of a new author, C. S. Forbes. Catherine Forbes holds a Ph.D. in Mathematical Statistics from The Ohio State University, USA, and is currently Senior Lecturer at Monash University, Victoria, Australia. Professor Merran Evans is currently Pro Vice-Chancellor, Planning and Quality at Monash University and obtained her Ph.D. in Econometrics from Monash University. Dr. Nicholas Hastings holds a Ph.D. in Operations Research from the University of Birmingham. Statistical Distributions 4th Edition

Formerly Mount Isa Mines Professor of Maintenance Engineering at Queensland University of Technology, Brisbane, Australia, he is currently Director and Consultant in physical asset management, Albany Interactive Pty Ltd. Dr. Brian Peacock has a background in ergonomics and industrial engineering which have provided a foundation for a long career in industry and academia, including 18 years in academia, 15 years with General Motors’ vehicle design and manufacturing organizations, and 4 years as discipline coordinating scientist for the National Space Biomedical Institute/NASA. He is a licensed professional engineer, a licensed private pilot, a certified professional ergonomist, and a fellow of both the Ergonomics and Human Factors Society (UK) and the Human Factors and Ergonomics Society (USA). He recently retired as a professor in the Department of Safety Science at Embry Riddle Aeronautical University, where he taught classes in system safety and applied ergonomics. x

The authors gratefully acknowledge the helpful suggestions and comments made by Harry Bartlett, Jim Conlan, Benoit Dulong, Alan Farley, Robert Kushler, Jerry W. Lewis, Allan T. Mense, Grant Reinman, and Dimitris Ververidis.

Introduction – Statistical Distributions 4th Edition

The number of puppies in a litter, the life of a light bulb, and the time to arrival of the next bus at a stop are all examples of random variables encountered in everyday life. Random variables have come to play an important role in nearly every field of study: in physics, chemistry, and engineering, and especially in the biological, social, and management sciences. Random variables are measured and analyzed in terms of their statistical and probabilistic properties, an underlying feature of which is the distribution function. Although the number of potential distribution models is very large, in practice a relatively small number have come to prominence, either because they have desirable mathematical characteristics or because they relate particularly well to some slice of reality or both. Statistical Distributions 4th Edition

This book gives a concise statement of leading facts relating to 40 distributions and includes diagrams so that shapes and other general properties may readily be appreciated. A consistent system of nomenclature is used throughout. We have found ourselves in need of just such a summary on frequent occasions—as students, as teachers, and as practitioners. This book has been prepared and revised in an attempt to fill the need for rapid access to information that must otherwise be gleaned from scattered and individually costly sources. In choosing the material, we have been guided by a utilitarian outlook. For example, some distributions that are special cases of more general families are given extended treatment where this is felt to be justified by applications. A general discussion of families or systems of distributions was considered beyond the scope of this book. In choosing the appropriate symbols and parameters for the description of each distribution, and especially where different but interrelated sets of symbols are in use in different fields, we have tried to strike a balance between the various usages, the need for a consistent system of nomenclature within the book, and typographic simplicity. We have given some methods of parameter estimation where we felt it was appropriate to do so. References listed in the Bibliography are not the primary sources but should be regarded as the first “port of call”. In addition to listing the properties of individual variates we have considered relationships between variates. This area is often obscure to the nonspecialist

Review – Statistical Distributions 4th Edition

“Overall, an excellent book for readers interested in qualitative data analysis. Highly recommended. Upper-division undergraduates through professionals.” (Choice, 1 October 2011)

“This new edition continues to illustrate the application of statistical methods to research across various disciplines, including medicine, engineering, business/finance, and the social sciences. Thoroughly revised and updated, the authors have refreshed this book to reflect the changes and current trends in statistical distribution theory that have occured since the publication of the previous edition eight years ago . . . key facts and formulas for forty major probability distributions are presented, making the book an ideal introduction to the general theory of statistical distributions as well as a quick reference on its basic principles”. (MyCFO, 22 December 2010)

“This new edition continues to illustrate the application of statistical methods to research across various disciplines, including medicine, engineering, business/finance, and the social sciences. Thoroughly revised and updated, the authors have refreshed this book to reflect the changes and current trends in statistical distribution theory that have occured since the publication of the previous edition eight years ago. The introductory chapters introduce the fundamental concepts of the distributions and the relationships between variables. For each distribution that follows, the key formulae, tables and diagrams are presented in a concise, user-friendly format. Key facts and formulas for forty major probability distributions are presented, making the book an ideal introduction to the general theory of statistical distributions as well as a quick reference on its basic principles”. (MyCFO, 22 December 2010)

From the Inside Flap

A new edition of the trusted guide on commonly used statistical distributions

Fully updated to reflect the latest developments on the topic, Statistical Distributions, Fourth Edition continues to serve as an authoritative guide on the application of statistical methods to research across various disciplines. The book provides a concise presentation of popular statistical distributions along with the necessary knowledge for their successful use in data modeling and analysis.

Following a basic introduction, forty popular distributions are outlined in individual chapters that are complete with related facts and formulas. Reflecting the latest changes and trends in statistical distribution theory, the Fourth Edition features:

  • A new chapter on queuing formulas that discusses standard formulas that often arise from simple queuing systems
  • Methods for extending independent modeling schemes to the dependent case, covering techniques for generating complex distributions from simple distributions
  • New coverage of conditional probability, including conditional expectations and joint and marginal distributions
  • Commonly used tables associated with the normal (Gaussian), student-t, F and chi-square distributions
  • Additional reviewing methods for the estimation of unknown parameters, such as the method of percentiles, the method of moments, maximum likelihood inference, and Bayesian inference

Statistical Distributions, Fourth Edition is an excellent supplement for upper-undergraduate and graduate level courses on the topic. It is also a valuable reference for researchers and practitioners in the fields of engineering, economics, operations research, and the social sciences who conduct statistical analyses.

About the author (2011)

CATHERINE FORBES, PhD, is Senior Lecturer in the Department of Econometrics and Business Statistics at Monash University (Australia). With experience as a statistical consultant for business, manufacturing industries, and social welfare agencies, her research focuses on the areas of Bayesian econometrics, financial modeling, time series analysis, and model selection.

MERRAN EVANS, PhD, is Professor and Pro Vice-Chancellor (Planning and Quality) at Monash University. Dr. Evans has over thirty years of academic experience in the fields of statistics and econometrics.

NICHOLAS HASTINGS, PhD, is Director and Principal Consultant in physical asset management at Albany Interactive Pty Ltd. Dr. Hastings has published extensively in the areas of engineering management and asset management.

BRIAN PEACOCK, PhD, is Founder of Brian Peacock Ergonomics (BPE) Pte. Ltd., a Singapore-based firm that offers ergonomics and human factors education, training, and consulting services. He has previously held consulting positions at General Motors Company and the National Space Biomedical Research Institute/NASA.

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