# Statistical Distributions, 4th Edition

ISBN: 978-0-470-39063-4
230 pages
December 2010

## Description

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.

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1 Introduction.

2 Terms and Symbols.

2.1 Probability, Random Variable, Variate and Number.

2.2 Range, Quantile, Probability and Domain.

2.3 Distribution Function and Survival Function.

2.4 Inverse Distribution and Inverse Survival Function.

2.5 Probability Density Function and Probability Function.

2.6 Other Associated Functions and Quantities.

3 General Variate Relationships.

3.1 Introduction.

3.2 Function of a Variate.

3.3 One-to-One Transformations and Inverses.

3.4 Variate Relationships Under One-to-One Transformation.

3.5 Parameters, Variate, and Function Notation.

3.6 Transformation of Location and Scale.

3.7 Transformation from the Rectangular Variate.

3.8 Many-to-One Transformations.

4 Multivariate Distributions.

4.1 Joint Distributions.

4.2 Marginal Distributions.

4.3 Independence.

4.4 Conditional Distributions.

4.5 Bayes' Theorem.

4.6 Functions of a Multivariate.

5 Stochastic Modeling.

5.1 Introduction.

5.2 Independent Variates.

5.3 Mixture Distributions.

5.4 Skew-Symmetric Distributions.

5.5 Conditional Skewness.

5.6 Dependent Variates.

6 Parameter Inference.

6.1 Introduction.

6.2 Method of Percentiles Estimation.

6.3 Method of Moments Estimation.

6.4 Maximum Likelihood Inference.

6.5 Bayesian Inference.

7 Bernoulli Distribution.

7.1 Random Number Generation.

7.2 Curtailed Bernoulli Trial Sequences.

7.3 Urn Sampling Scheme.

7.4 Note.

8 Beta Distribution.

8.1 Notes on Beta and Gamma Functions.

8.2 Variate Relationships.

8.3 Parameter Estimation.

8.4 Random Number Generation.

8.5 Inverted Beta Distribution.

8.6 Noncentral Beta Distribution.

8.7 Beta Binomial Distribution.

9 Binomial Distribution.

9.1 Variate Relationships.

9.2 Parameter Estimation.

9.3 Random Number Generation.

10 Cauchy Distribution.

10.1 Note.

10.2 Variate Relationships.

10.3 Random Number Generation.

10.4 Generalized Form.

11 Chi-Squared Distribution.

11.1 Variate Relationships.

11.2 Random Number Generation.

11.3 Chi Distribution.

12 Chi-Squared (Noncentral) Distribution.

12.1 Variate Relationships.

13 Dirichlet Distribution.

13.1 Variate Relationships.

13.2 Dirichlet Multinomial Distribution.

14 Empirical Distribution Function.

14.1 Estimation from Uncensored Data.

14.2 Estimation from Censored Data.

14.3 Parameter Estimation.

14.4 Example.

14.5 Graphical Method for the Modified Order-Numbers.

14.6 Model Accuracy.

15 Erlang Distribution.

15.1 Variate Relationships.

15.2 Parameter Estimation.

15.3 Random Number Generation.

16 Error Distribution.

16.1 Note.

16.2 Variate Relationships.

17 Exponential Distribution.

17.1 Note.

17.2 Variate Relationships.

17.3 Parameter Estimation.

17.4 Random Number Generation.

18 Exponential Family.

18.1 Members of the Exponential Family.

18.2 Univariate One-Parameter Exponential Family.

18.3 Estimation.

18.4 Generalized Exponential Distributions.

19 Extreme Value (Gumbel) Distribution.

19.1 Note.

19.2 Variate Relationships.

19.3 Parameter Estimation.

19.4 Random Number Generation.

20 F (Variance Ratio) or Fisher{ Snedecor Distribution.

20.1 Variate Relationships.

21 F (Noncentral) Distribution.

21.1 Variate Relationships.

22 Gamma Distribution.

22.1 Variate Relationships.

22.2 Parameter Estimation.

22.3 Random Number Generation.

22.4 Inverted Gamma Distribution.

22.5 Normal Gamma Distribution.

22.6 Generalized Gamma Distribution.

22.6.1 Variate Relationships.

23 Geometric Distribution.

23.1 Notes.

23.2 Variate Relationships.

23.3 Random Number Generation.

24 Hypergeometric Distribution.

24.1 Note.

24.2 Variate Relationships.

24.3 Parameter Estimation.

24.4 Random Number Generation.

24.5 Negative Hypergeometric Distribution.

24.6 Generalized Hypergeometric (Series) Distribution.

25 Inverse Gaussian (Wald) Distribution.

25.1 Variate Relationships.

25.2 Parameter Estimation.

26 Laplace Distribution.

26.1 Variate Relationships.

26.2 Parameter Estimation.

26.3 Random Number Generation.

27 Logarithmic Series Distribution.

27.1 Variate Relationships.

27.2 Parameter Estimation.

28 Logistic Distribution.

28.1 Notes.

28.2 Variate Relationships.

28.3 Parameter Estimation.

28.4 Random Number Generation.

29 Lognormal Distribution.

29.1 Variate Relationships.

29.2 Parameter Estimation.

29.3 Random Number Generation.

30 Multinomial Distribution.

30.1 Variate Relationships.

30.2 Parameter Estimation.

31 Multivariate Normal (Multinormal) Distribution.

31.1 Variate Relationships.

31.2 Parameter Estimation.

32 Negative Binomial Distribution.

32.1 Note.

32.2 Variate Relationships.

32.3 Parameter Estimation.

32.4 Random Number Generation.

33 Normal (Gaussian) Distribution.

33.1 Variate Relationships.

33.2 Parameter Estimation.

33.3 Random Number Generation.

33.4 Truncated Normal Distribution.

33.5 Variate Relationships.

34 Pareto Distribution.

34.1 Note.

34.2 Variate Relationships.

34.3 Parameter Estimation.

34.4 Random Number Generation.

35 Poisson Distribution.

35.1 Note.

35.2 Variate Relationships.

35.3 Parameter Estimation.

35.4 Random Number Generation.

36 Power Function Distribution.

36.1 Variate Relationships.

36.2 Parameter Estimation.

36.3 Random Number Generation.

37 Power Series (Discrete) Distribution.

37.1 Note.

37.2 Variate Relationships.

37.3 Parameter Estimation.

38 Queuing Formulas.

38.1 Characteristics of Queuing Systems.

38.2 Definitions, Notation and Terminology.

38.3 General Formulas.

38.4 Some Standard Queuing Systems.

39 Rayleigh Distribution.

39.1 Variate Relationships.

39.2 Parameter Estimation.

40 Rectangular (Uniform) Continuous Distribution.

40.1Variate Relationships.

40.2 Parameter Estimation.

40.3 Random Number Generation.

41 Rectangular (Uniform) Discrete Distribution.

41.1 General Form.

41.2 Parameter Estimation.

42 Student's t Distribution.

42.1 Variate Relationships.

42.2 Random Number Generation.

43 Student's t (Noncentral) Distribution.

43.1 Variate Relationships.

44 Triangular Distribution.

44.1 Variate Relationships.

44.2 Random Number Generation.

45 von Mises Distribution.

45.1 Note.

45.2 Variate Relationships.

45.3 Parameter Estimation.

46 Weibull Distribution.

46.1 Note.

46.2 Variate Relationships.

46.3 Parameter Estimation.

46.4 Random Number Generation.

46.5 Three-Parameter Weibull Distribution.

46.6Three-Parameter Weibull Random Number Generation.

46.7 Bi-Weibull Distribution.

46.8 Five-Parameter Bi-Weibull Distribution.

Bi-Weibull Random Number Generation.

Bi-Weibull Graphs.

46.9 Weibull Family.

47 Wishart (Central) Distribution.

47.1 Note.

47.2 Variate Relationships.

48 Statistical Tables.

Bibliography.

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## Author Information

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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## Reviews

"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)

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