statistical distributions, 4th ed. - editorial dinosaurio · pdf filecontents preface xvii 1....

Statistical Distributions

Statistical DistributionsFourth Edition
Catherine ForbesMonash University, Victoria, Australia
Merran EvansMonash University, Victoria, Australia
Nicholas HastingsAlbany Interactive, Victoria, Australia
Brian PeacockBrian Peacock Ergonomics, SIM University, Singapore

Copyright 2011 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No 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., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax(978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission shouldbe addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts inpreparing this book, they make no representations or warranties with respect to the accuracy orcompleteness of the contents of this book and specifically disclaim any implied warranties ofmerchantability or fitness for a particular purpose. No warranty may be created or extended by salesrepresentatives or written sales materials. The advice and strategies contained herein may not be suitablefor your situation. You should consult with a professional where appropriate. Neither the publisher norauthor shall be liable for any loss of profit or any other commercial damages, including but not limited tospecial, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact ourCustomer Care Department within the United States at (800) 762-2974, outside the United States at (317)572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print maynot be available in electronic formats. For more information about Wiley products, visit our web site atwww.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Statistical distributions. 4th ed. / Catherine Forbes . . . [et al.].p. cm. (Wiley series in probability and statistics)
Includes bibliographical references and index.ISBN 978-0-470-39063-4 (pbk.)
1. Distribution (Probability theory) I. Forbes, Catherine.QA273.6.E92 2010519.24dc22
2009052131
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1

TOJeremy and Elana ForbesCaitlin and Eamon Evans
Tina HastingsEileen Peacock

Contents
Preface xvii
1. Introduction 1
2. Terms and Symbols 3
2.1 Probability, Random Variable, Variate, and Number 3Probabilistic Experiment 3Sample Space 3Random Variable 3Variate 3Random Number 4
2.2 Range, Quantile, Probability Statement, and Domain 4Range 4Quantile 5Probability Statement 5Probability Domain 5
2.3 Distribution Function and Survival Function 5Distribution Function 5Survival Function 6
2.4 Inverse Distribution Function and Inverse Survival Function 7Inverse Survival Function 8
2.5 Probability Density Function and Probability Function 82.6 Other Associated Functions and Quantities 9
3. General Variate Relationships 15
3.1 Introduction 153.2 Function of a Variate 153.3 One-to-One Transformations and Inverses 16
Inverse of a One-to-One Function 173.4 Variate Relationships Under One-to-One Transformation 17
Probability Statements 17Distribution Function 17Inverse Distribution Function 18Equivalence of Variates 18Inverse Function of a Variate 18
vii

viii Contents
3.5 Parameters, Variate, and Function Notation 19Variate and Function Notation 19
3.6 Transformation of Location and Scale 203.7 Transformation from the Rectangular Variate 203.8 Many-to-One Transformations 22
Symmetrical Distributions 22
4. Multivariate Distributions 24
4.1 Joint Distributions 24Joint Range 24Bivariate Quantile 24Joint Probability Statement 24Joint Probability Domain 25Joint Distribution Function 25Joint Probability Density Function 25Joint Probability Function 25
4.2 Marginal Distributions 26Marginal Probability Density Function and MarginalProbability Function 26
4.3 Independence 274.4 Conditional Distributions 28
Conditional Probability Function and ConditionalProbability Density Function 28Composition 29
4.5 Bayes Theorem 304.6 Functions of a Multivariate 30
5. Stochastic Modeling 32
5.1 Introduction 325.2 Independent Variates 325.3 Mixture Distributions 33
Finite Mixture 33Infinite Mixture of Distributions 35
5.4 Skew-Symmetric Distributions 385.5 Distributions Characterized by Conditional Skewness 395.6 Dependent Variates 42
6. Parameter Inference 44
6.1 Introduction 446.2 Method of Percentiles Estimation 446.3 Method of Moments Estimation 456.4 Maximum Likelihood Inference 47
Properties of MLEs 47Approximate Sampling Distribution for Fixed n 48

Contents ix
6.5 Bayesian Inference 50Marginal Posteriors 51
7. Bernoulli Distribution 53
7.1 Random Number Generation 537.2 Curtailed Bernoulli Trial Sequences 537.3 Urn Sampling Scheme 547.4 Note 54
8. Beta Distribution 55
8.1 Notes on Beta and Gamma Functions 56Definitions 56Interrelationships 56Special Values 57Alternative Expressions 57
8.2 Variate Relationships 578.3 Parameter Estimation 598.4 Random Number Generation 608.5 Inverted Beta Distribution 608.6 Noncentral Beta Distribution 618.7 Beta Binomial Distribution 61
9. Binomial Distribution 62
9.1 Variate Relationships 649.2 Parameter Estimation 659.3 Random Number Generation 65
10. Cauchy Distribution 66
10.1 Note 6610.2 Variate Relationships 6710.3 Random Number Generation 6810.4 Generalized Form 68
11. Chi-Squared Distribution 69
11.1 Variate Relationships 7111.2 Random Number Generation 7211.3 Chi Distribution 73
12. Chi-Squared (Noncentral) Distribution 74
12.1 Variate Relationships 75

x Contents
13. Dirichlet Distribution 77
13.1 Variate Relationships 7713.2 Dirichlet Multinomial Distribution 78
14. Empirical Distribution Function 79
14.1 Estimation from Uncensored Data 7914.2 Estimation from Censored Data 7914.3 Parameter Estimation 8114.4 Example 8114.5 Graphical Method for the Modified Order-Numbers 8114.6 Model Accuracy 83
15. Erlang Distribution 84
16. Error Distribution 86
16.1 Note 8716.2 Variate Relationships 87
17. Exponential Distribution 88
17.1 Note 8917.2 Variate Relationships 9117.3 Parameter Estimation 9217.4 Random Number Generation 92
18. Exponential Family 93
18.1 Members of the Exponential Family 9318.2 Univariate One-Parameter Exponential Family 9318.3 Parameter Estimation 9518.4 Generalized Exponential Distributions 95
Generalized Students t Distribution 95Variate Relationships 96Generalized Exponential Normal Distribution 96Generalized Lognormal Distribution 96Variate Relationships 97

Contents xi
19. Extreme Value (Gumbel) Distribution 98
20. F (Variance Ratio) or FisherSnedecor Distribution 102
21. F (Noncentral) Distribution 107
22. Gamma Distribution 109
22.1 Variate Relationships 11022.2 Parameter Estimation 11122.3 Random Number Generation 11222.4 Inverted Gamma Distribution 11222.5 Normal Gamma Distribution 11222.6 Generalized Gamma Distribution 113
Variate Relationships 113
23. Geometric Distribution 114
23.1 Notes 11523.2 Variate Relationships 11523.3 Random Number Generation 116
24. Hypergeometric Distribution 117
24.1 Note 11824.2 Variate Relationships 11824.3 Parameter Estimation 11824.4 Random Number Generation 11924.5 Negative Hypergeometric Distribution 11924.6 Generalized Hypergeometric Distribution 119
25. Inverse Gaussian (Wald) Distribution 120
25.1 Variate Relationships 12125.2 Parameter Estimation 121

xii Contents
26. Laplace Distribution 122
27. Logarithmic Series Distribution 125
28. Logistic Distribution 127
28.1 Notes 12828.2 Variate Relationships 12828.3 Parameter Estimation 13028.4 Random Number Generation 130
29. Lognormal Distribution 131
30. Multinomial Distribution 135
31. Multivariate Normal (Multinormal) Distribution 137
32. Negative Binomial Distribution 139
33. Normal (Gaussian) Distribution 143

statistical distributions, 4th ed. - editorial dinosaurio · pdf filecontents preface xvii 1....

Documents