Probability and Statistics

[Pages:911]Probability and Statistics

Fourth Edition

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Probability and Statistics

Fourth Edition

Morris H. DeGroot

Carnegie Mellon University

Mark J. Schervish

Carnegie Mellon University

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Library of Congress Cataloging-in-Publication Data

DeGroot, Morris H., 1931?1989. Probability and statistics / Morris H. DeGroot, Mark J. Schervish.--4th ed. p. cm. ISBN 978-0-321-50046-5 1. Probabilities--Textbooks. 2. Mathematical statistics--Textbooks.

I. Schervish, Mark J. II. Title. QA273.D35 2012 519.2--dc22 2010001486

Copyright ? 2012, 2002 Pearson Education, Inc.

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ISBN 10: 0-321-50046-6 ISBN 13: 978-0-321-50046-5

To the memory of Morrie DeGroot. MJS

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Contents

Preface xi

1 Introduction to Probability 1

1.1 The History of Probability 1 1.2 Interpretations of Probability 2 1.3 Experiments and Events 5 1.4 Set Theory 6 1.5 The Definition of Probability 16 1.6 Finite Sample Spaces 22 1.7 Counting Methods 25 1.8 Combinatorial Methods 32 1.9 Multinomial Coefficients 42 1.10 The Probability of a Union of Events 46 1.11 Statistical Swindles 51 1.12 Supplementary Exercises 53

2 Conditional Probability 55

2.1 The Definition of Conditional Probability 55 2.2 Independent Events 66 2.3 Bayes' Theorem 76 2.4 The Gambler's Ruin Problem 86 2.5 Supplementary Exercises 90

3 Random Variables and Distributions 93

3.1 Random Variables and Discrete Distributions 93 3.2 Continuous Distributions 100 3.3 The Cumulative Distribution Function 107 3.4 Bivariate Distributions 118 3.5 Marginal Distributions 130 3.6 Conditional Distributions 141 3.7 Multivariate Distributions 152 3.8 Functions of a Random Variable 167 3.9 Functions of Two or More Random Variables 175 3.10 Markov Chains 188 3.11 Supplementary Exercises 202

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