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Statistical Distributions


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Table of Contents

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.

About the Author

CATHERINE FORBES, PhD, is Senior Lecturer in the Departmentof Econometrics and Business Statistics at Monash University(Australia). With experience as a statistical consultant forbusiness, 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 overthirty years of academic experience in the fields of statistics andeconometrics. NICHOLAS HASTINGS, PhD, is Director and PrincipalConsultant in physical asset management at Albany Interactive PtyLtd. Dr. Hastings has published extensively in the areas ofengineering management and asset management. BRIAN PEACOCK, PhD, is Founder of Brian PeacockErgonomics (BPE) Pte. Ltd., a Singapore-based firm that offersergonomics and human factors education, training, and consultingservices. He has previously held consulting positions at GeneralMotors Company and the National Space Biomedical ResearchInstitute/NASA.


"Overall, an excellent book for readers interested in qualitativedata analysis. Highly recommended. Upper-division undergraduatesthrough professionals." (Choice, 1 October 2011) "This new edition continues to illustrate the application ofstatistical methods to research across various disciplines,including medicine, engineering, business/finance, and the socialsciences. Thoroughly revised and updated, the authors haverefreshed this book to reflect the changes and current trends instatistical distribution theory that have occured since thepublication of the previous edition eight years ago ... key factsand formulas for forty major probability distributions arepresented, making the book an ideal introduction to the generaltheory of statistical distributions as well as a quick reference onits basic principles". (MyCFO, 22 December 2010) "This new edition continues to illustrate the application ofstatistical methods to research across various disciplines,including medicine, engineering, business/finance, and the socialsciences. Thoroughly revised and updated, the authors haverefreshed this book to reflect the changes and current trends instatistical distribution theory that have occured since thepublication of the previous edition eight years ago. Theintroductory chapters introduce the fundamental concepts of thedistributions and the relationships between variables. For eachdistribution that follows, the key formulae, tables and diagramsare presented in a concise, user-friendly format. Key facts andformulas for forty major probability distributions are presented,making the book an ideal introduction to the general theory ofstatistical distributions as well as a quick reference on its basicprinciples". (MyCFO, 22 December 2010)

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