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A First Course in Probability(Ross).pdf

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    Cover
    Title Page
    Copyright
    Contents
    Preface
    1 Combinatorial Analysis
    1.1 Introduction
    1.2 The Basic Principle of Counting
    1.3 Permutations
    1.4 Combinations
    1.5 Multinomial Coefficients
    1.6 The Number of Integer Solutions of Equations
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    2 Axioms of Probability
    2.1 Introduction
    2.2 Sample Space and Events
    2.3 Axioms of Probability
    2.4 Some Simple Propositions
    2.5 Sample Spaces Having Equally Likely Outcomes
    2.6 Probability as a Continuous Set Function
    2.7 Probability as a Measure of Belief
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    3 Conditional Probability and Independence
    3.1 Introduction
    3.2 Conditional Probabilities
    3.3 Bayes's Formula
    3.4 Independent Events
    3.5 P(·|F) Is a Probability
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    4 Random Variables
    4.1 Random Variables
    4.2 Discrete Random Variables
    4.3 Expected Value
    4.4 Expectation of a Function of a Random Variable
    4.5 Variance
    4.6 The Bernoulli and Binomial Random Variables
    4.6.1 Properties of Binomial Random Variables
    4.6.2 Computing the Binomial Distribution Function
    4.7 The Poisson Random Variable
    4.7.1 Computing the Poisson Distribution Function
    4.8 Other Discrete Probability Distributions
    4.8.1 The Geometric Random Variable
    4.8.2 The Negative Binomial Random Variable
    4.8.3 The Hypergeometric Random Variable
    4.8.4 The Zeta (or Zipf) Distribution
    4.9 Expected Value of Sums of Random Variables
    4.10 Properties of the Cumulative Distribution Function
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    5 Continuous Random Variables
    5.1 Introduction
    5.2 Expectation and Variance of Continuous Random Variables
    5.3 The Uniform Random Variable
    5.4 Normal Random Variables
    5.4.1 The Normal Approximation to the Binomial Distribution
    5.5 Exponential Random Variables
    5.5.1 Hazard Rate Functions
    5.6 Other Continuous Distributions
    5.6.1 The Gamma Distribution
    5.6.2 The Weibull Distribution
    5.6.3 The Cauchy Distribution
    5.6.4 The Beta Distribution
    5.7 The Distribution of a Function of a Random Variable
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    6 Jointly Distributed Random Variables
    6.1 Joint Distribution Functions
    6.2 Independent Random Variables
    6.3 Sums of Independent Random Variables
    6.3.1 Identically Distributed Uniform Random Variables
    6.3.2 Gamma Random Variables
    6.3.3 Normal Random Variables
    6.3.4 Poisson and Binomial Random Variables
    6.3.5 Geometric Random Variables
    6.4 Conditional Distributions: Discrete Case
    6.5 Conditional Distributions: Continuous Case
    6.6 Order Statistics
    6.7 Joint Probability Distribution of Functions of Random Variables
    6.8 Exchangeable Random Variables
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    7 Properties of Expectation
    7.1 Introduction
    7.2 Expectation of Sums of Random Variables
    7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method
    7.2.2 The Maximum–Minimums Identity
    7.3 Moments of the Number of Events that Occur
    7.4 Covariance, Variance of Sums, and Correlations
    7.5 Conditional Expectation
    7.5.1 Definitions
    7.5.2 Computing Expectations by Conditioning
    7.5.3 Computing Probabilities by Conditioning
    7.5.4 Conditional Variance
    7.6 Conditional Expectation and Prediction
    7.7 Moment Generating Functions
    7.7.1 Joint Moment Generating Functions
    7.8 Additional Properties of Normal Random Variables
    7.8.1 The Multivariate Normal Distribution
    7.8.2 The Joint Distribution of the Sample Mean and Sample Variance
    7.9 General Definition of Expectation
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    8 Limit Theorems
    8.1 Introduction
    8.2 Chebyshev's Inequality and the Weak Law of Large Numbers
    8.3 The Central Limit Theorem
    8.4 The Strong Law of Large Numbers
    8.5 Other Inequalities
    8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable
    Summary
    Problems
    Theoretical Exercises
    Self-Test Problems and Exercises
    9 Additional Topics in Probability
    9.1 The Poisson Process
    9.2 Markov Chains
    9.3 Surprise, Uncertainty, and Entropy
    9.4 Coding Theory and Entropy
    Summary
    Problems and Theoretical Exercises
    Self-Test Problems and Exercises
    References
    10 Simulation
    10.1 Introduction
    10.2 General Techniques for Simulating Continuous Random Variables
    10.2.1 The Inverse Transformation Method
    10.2.2 The Rejection Method
    10.3 Simulating from Discrete Distributions
    10.4 Variance Reduction Techniques
    10.4.1 Use of Antithetic Variables
    10.4.2 Variance Reduction by Conditioning
    10.4.3 Control Variates
    Summary
    Problems
    Self-Test Problems and Exercises
    Reference
    Answers to Selected Problems
    Solutions to Self-Test Problems and Exercises
    Index
    A
    B
    C
    D
    E
    F
    G
    H
    I
    J
    K
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    Z
    A FIRST COURSE IN PROBABILITY
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    A FIRST COURSE IN PROBABILITY Eighth Edition Sheldon Ross University of Southern California Upper Saddle River, New Jersey 07458
    Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. A first course in probability / Sheldon Ross. — 8th ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X 1. Probabilities—Textbooks. QA273.R83 2010 519.2—dc22 I. Title. 2008033720 Editor in Chief, Mathematics and Statistics: Deirdre Lynch Senior Project Editor: Rachel S. Reeve Assistant Editor: Christina Lepre Editorial Assistant: Dana Jones Project Manager: Robert S. Merenoff Associate Managing Editor: Bayani Mendoza de Leon Senior Managing Editor: Linda Mihatov Behrens Senior Operations Supervisor: Diane Peirano Marketing Assistant: Kathleen DeChavez Creative Director: Jayne Conte Art Director/Designer: Bruce Kenselaar AV Project Manager: Thomas Benfatti Compositor: Integra Software Services Pvt. Ltd, Pondicherry, India Cover Image Credit: Getty Images, Inc. © 2010, 2006, 2002, 1998, 1994, 1988, 1984, 1976 by Pearson Education, Inc., Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X Pearson Education, Ltd., London Pearson Education Australia PTY. Limited, Sydney Pearson Education Singapore, Pte. Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educaci ´on de Mexico, S.A. de C.V. Pearson Education – Japan, Tokyo Pearson Education Malaysia, Pte. Ltd Pearson Education Upper Saddle River, New Jersey
    For Rebecca
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    Contents Preface 1 Combinatorial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . 1.1 1.2 The Basic Principle of Counting . 1.3 Permutations . 1.4 Combinations 1.5 Multinomial Coefficients . 1.6 The Number of Integer Solutions of Equations . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . Problems . . Theoretical Exercises . . . . . Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Axioms of Probability . Introduction . . . . . 2.1 . . . . . . . 2.2 Sample Space and Events . . . . . . . . . . 2.3 Axioms of Probability . . . . . . . 2.4 Some Simple Propositions . . . . . . . . 2.5 Sample Spaces Having Equally Likely Outcomes . . . 2.6 Probability as a Continuous Set Function . . . . . 2.7 Probability as a Measure of Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . Problems . . Theoretical Exercises . . . . . Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 1 1 3 5 9 12 15 16 18 20 22 22 22 26 29 33 44 48 49 50 54 56 3 Conditional Probability and Independence . . . . . . . . . . . . . . . . . . . . . . . . Introduction . 3.1 3.2 Conditional Probabilities . 3.3 Bayes’s Formula . . . . 3.4 3.5 P(·|F) Is a Probability . . . . . . . . . . . . . Independent Events . . . . . . Summary . . Problems . . Theoretical Exercises . . . . . Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 58 . . . . . . 58 . . . . 65 . . . . . . 79 . . . . . . 93 . . . . 101 . . . . . 102 . . . . 110 . . . . 114 4 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Random Variables . . . 4.2 Discrete Random Variables 4.3 Expected Value . . . . . . . . . . . 4.4 Expectation of a Function of a Random Variable . . . 4.5 Variance . . 4.6 The Bernoulli and Binomial Random Variables . . . . 4.6.1 Properties of Binomial Random Variables . . . 4.6.2 Computing the Binomial Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 . . . . . . 117 . . . . 123 . . . 125 . . . . 128 . . . . 132 . . . . 134 . . . . 139 . . 142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
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