# Download 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.

Table of Contents

## Content – Statistical Distributions 4th Edition

Preface xvii

- Introduction 1
- 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 - 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

- 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 - 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 - 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

- 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 - 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 - Binomial Distribution 62

9.1 Variate Relationships 64

9.2 Parameter Estimation 65

9.3 Random Number Generation 65 - Cauchy Distribution 66

10.1 Note 66

10.2 Variate Relationships 67

10.3 Random Number Generation 68

10.4 Generalized Form 68 - Chi-Squared Distribution 69

11.1 Variate Relationships 71

11.2 Random Number Generation 72

11.3 Chi Distribution 73 - Chi-Squared (Noncentral) Distribution 74

12.1 Variate Relationships 75

Statistical Distributions 4th Edition

- Dirichlet Distribution 77
- 13.1 Variate Relationships 77
- 13.2 Dirichlet Multinomial Distribution 78
- 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
- Erlang Distribution 84
- 15.1 Variate Relationships 85
- 15.2 Parameter Estimation 85
- 15.3 Random Number Generation 85
- Error Distribution 86
- 16.1 Note 87
- 16.2 Variate Relationships 87
- Exponential Distribution 88
- 17.1 Note 89
- 17.2 Variate Relationships 91
- 17.3 Parameter Estimation 92
- 17.4 Random Number Generation 92
- 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 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
- F (Variance Ratio) or Fisher–Snedecor Distribution 102
- 20.1 Variate Relationships 103
- F (Noncentral) Distribution 107
- 21.1 Variate Relationships 108
- 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
- Geometric Distribution 114
- 23.1 Notes 115
- 23.2 Variate Relationships 115
- 23.3 Random Number Generation 116
- 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
- Inverse Gaussian (Wald) Distribution 120
- 25.1 Variate Relationships 121
- 25.2 Parameter Estimation 121 Laplace Distribution 122
- 26.1 Variate Relationships 124
- 26.2 Parameter Estimation 124
- 26.3 Random Number Generation 124
- Logarithmic Series Distribution 125
- 27.1 Variate Relationships 126
- 27.2 Parameter Estimation 126
- Logistic Distribution 127
- 28.1 Notes 128
- 28.2 Variate Relationships 128
- 28.3 Parameter Estimation 130
- 28.4 Random Number Generation 130
- Lognormal Distribution 131
- 29.1 Variate Relationships 132
- 29.2 Parameter Estimation 134
- 29.3 Random Number Generation 134
- Multinomial Distribution 135
- 30.1 Variate Relationships 136
- 30.2 Parameter Estimation 136
- Multivariate Normal (Multinormal) Distribution 137
- 31.1 Variate Relationships 138
- 31.2 Parameter Estimation 138
- Negative Binomial Distribution 139
- 32.1 Note 140
- 32.2 Variate Relationships 141
- 32.3 Parameter Estimation 142
- 32.4 Random Number Generation 142
- Normal (Gaussian) Distribution 143
- 33.1 Variate Relationships 144
- 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

- Pareto Distribution 149

34.1 Note 149

34.2 Variate Relationships 150

34.3 Parameter Estimation 151

34.4 Random Number Generation 151 - Poisson Distribution 152

35.1 Note 153

35.2 Variate Relationships 153

35.3 Parameter Estimation 156

35.4 Random Number Generation 156 - Power Function Distribution 157

36.1 Variate Relationships 157

36.2 Parameter Estimation 159

36.3 Random Number Generation 159 - Power Series (Discrete) Distribution 160

37.1 Note 160

37.2 Variate Relationships 161

37.3 Parameter Estimation 161 - 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

- Rayleigh Distribution 173 Statistical Distributions 4th Edition

39.1 Variate Relationships 173

39.2 Parameter Estimation 175 - Rectangular (Uniform) Continuous Distribution 176

40.1 Variate Relationships 177

40.2 Parameter Estimation 179

40.3 Random Number Generation 179 - Rectangular (Uniform) Discrete Distribution 180

41.1 General Form 181

41.2 Parameter Estimation 182 - Student’s t Distribution 183

42.1 Variate Relationships 185

42.2 Random Number Generation 186 - Student’s t (Noncentral) Distribution 187

43.1 Variate Relationships 188 - Triangular Distribution 189

44.1 Variate Relationships 189

44.2 Random Number Generation 190 - von Mises Distribution 191

45.1 Note 191

45.2 Variate Relationships 192

45.3 Parameter Estimation 192 - 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

- Wishart (Central) Distribution 202

47.1 Note 203

47.2 Variate Relationships 203 - 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.

**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.