Statistical Distributions, 4th ed.

1 Statistical DistributionsStatistical DistributionsFourth EditionCatherine ForbesMonash University, Victoria, AustraliaMerran EvansMonash University, Victoria, AustraliaNicholas HastingsAlbany Interactive, Victoria, AustraliaBrian PeacockBrian Peacock Ergonomics, SIM University, SingaporeCopyright 2011 by John Wiley & Sons, Inc. All rights by John Wiley & Sons, Inc., Hoboken, New simultaneously in part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee tothe Copyright Clearance Center, Inc.

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4 [et al.].p. cm. (Wiley series in probability and statistics)Includes bibliographical references and 978-0-470-39063-4 (pbk.)1. distribution (Probability theory) I. Forbes, 4 dc222009052131 Printed in the United States of and Elana ForbesCaitlin and Eamon EvansTina HastingsEileen PeacockContentsPrefacexvii1. Introduction12. Terms and , random Variable, Variate, and Number3 Probabilistic Experiment3 Sample Space3 random Variable3 Variate3 random , Quantile, Probability Statement, and Domain4 Range4 Quantile5 Probability Statement5 Probability Function and Survival Function5 distribution Function5 Survival distribution Function and Inverse Survival Function7 Inverse Survival Density Function and Probability Associated Functions and Quantities93.

5 General Variate of a Transformations and Inverses16 Inverse of a One-to-One Relationships Under One-to-One Transformation17 Probability Statements17 distribution Function17 Inverse distribution Function18 Equivalence of Variates18 Inverse Function of a , Variate, and Function Notation19 Variate and Function of Location and from the Rectangular Transformations22 Symmetrical Distributions224. Multivariate Distributions24 Joint Range24 Bivariate Quantile24 Joint Probability Statement24 Joint Probability Domain25 Joint distribution Function25 Joint Probability Density Function25 Joint Probability Distributions26 Marginal Probability Density Function and MarginalProbability Distributions28 Conditional Probability Function and ConditionalProbability Density of a Multivariate305.

6 Stochastic Distributions33 Finite Mixture33 Infinite Mixture of Characterized by Conditional Variates426. Parameter of Percentiles of Moments Likelihood Inference47 properties of MLEs47 Approximate Sampling distribution for Inference50 Marginal Posteriors517. Bernoulli Number Bernoulli Trial Sampling Beta on Beta and Gamma Functions56 Definitions56 Interrelationships56 Special Values57 Alternative Number Beta Beta Binomial Distribution619. Binomial Number Generation6510. Cauchy Variate random Number Generalized Form6811. Chi-Squared Variate random Number Chi Distribution7312. Chi-Squared (Noncentral) Variate Relationships75xContents13. Dirichlet Variate Dirichlet Multinomial Distribution7814.

7 Empirical distribution Estimation from Uncensored Estimation from Censored Parameter Graphical Method for the Modified Model Accuracy8315. Erlang Variate Parameter random Number Generation8516. Error Variate Relationships8717. Exponential Variate Parameter random Number Generation9218. Exponential Members of the Exponential Univariate One-Parameter Exponential Parameter Generalized Exponential Distributions95 Generalized Student stDistribution95 Variate Relationships96 Generalized Exponential Normal Distribution96 Generalized Lognormal Distribution96 Variate Relationships97 Contentsxi19. Extreme Value (Gumbel) Variate Parameter random Number (Variance Ratio) or Fisher Snedecor Variate (Noncentral) Variate Relationships10822.

8 Gamma Variate Parameter random Number Inverted Gamma Normal Gamma Generalized Gamma Distribution113 Variate Relationships11323. Geometric Variate random Number Generation11624. Hypergeometric Variate Parameter random Number Negative Hypergeometric Generalized Hypergeometric Distribution11925. Inverse Gaussian (Wald) Variate Parameter Estimation121xiiContents26. Laplace Variate Parameter random Number Generation12427. Logarithmic Series Variate Parameter Estimation12628. Logistic Variate Parameter random Number Generation13029. Lognormal Variate Parameter random Number Generation13430. Multinomial Variate Parameter Estimation13631.

9 Multivariate Normal (Multinormal) Variate Parameter Estimation13832. Negative Binomial Variate Parameter random Number Generation14233. Normal (Gaussian) Variate Parameter random Number Truncated Normal Variate Relationships14834. Pareto Variate Parameter random Number Generation15135. Poisson Variate Parameter random Number Generation15636. Power Function Variate Parameter random Number Generation15937. Power Series (Discrete) Variate Parameter Estimation16138. Queuing Characteristics of Queuing Systems and Kendall-Lee Notation162 Characteristics of Queuing Systems162 Kendall-Lee Definitions, Notation, and Terminology164 Steady State164 Traffic Intensity and Traffic Density164 Notation and General Some Standard Queuing Systems166 TheM/M/1/G/ / System166 TheM/M/s/G/ / System166 TheM/G/1/G/ / System (Pollaczek-Khinchin)168 TheM/M/1/G/m/ System168 TheM/G/m/G/m/ System (Erlang)169xivContentsTheM/M/1/G/N/NSyst em (One Server, Finite PopulationN)169 TheM/M/s/G/N/NSystem (sServers, Finite PopulationN)17139.

10 Rayleigh Variate Parameter Estimation17540. Rectangular (Uniform) Continuous Variate Parameter random Number Generation17941. Rectangular (Uniform) Discrete General Parameter Estimation18242. Student Variate random Number Generation18643. Student st(Noncentral) Variate Relationships18844. Triangular Variate random Number Generation19045. von Mises Variate Parameter Estimation19246. Weibull Variate Parameter random Number Three-Parameter Weibull Three-Parameter Weibull random Number Bi-Weibull Five-Parameter Bi-Weibull Distribution198Bi-Weibull random Number Generation200Bi-Weibull Weibull Family20147. Wishart (Central) Variate Relationships20348.