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Mathematical Statistics with Applications 英文版.pdf

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    Front Cover
    Title Page
    Copyright
    Contents
    Preface
    Note to the Student
    1 What Is Statistics?
    1.1 Introduction
    Exercises
    1.2 Characterizing a Set of Measurements: Graphical Methods
    Exercises
    1.3 Characterizing a Set of Measurements: Numerical Methods
    Exercises
    1.4 How Inferences Are Made
    1.5 Theory and Reality
    1.6 Summary
    References and Further Readings
    Supplementary Exercises
    2 Probability
    2.1 Introduction
    2.2 Probability and Inference
    2.3 A Review of Set Notation
    Exercises
    2.4 A Probabilistic Model for an Experiment: The Discrete Case
    Exercises
    2.5 Calculating the Probability of an Event: The Sample-Point Method
    Exercises
    2.6 Tools for Counting Sample Points
    Exercises
    2.7 Conditional Probability and the Independence of Events
    Exercises
    2.8 Two Laws of Probability
    Exercises
    2.9 Calculating the Probability of an Event: The Event-Composition Method
    Exercises
    2.10 The Law of Total Probability and Bayes’ Rule
    Exercises
    2.11 Numerical Events and Random Variables
    Exercises
    2.12 Random Sampling
    2.13 Summary
    References and Further Readings
    Supplementary Exercises
    3 Discrete Random Variables and Their Probability Distributions
    3.1 Basic Definition
    3.2 The Probability Distribution for a Discrete Random Variable
    Exercises
    3.3 The Expected Value of a Random Variable or a Function of a Random Variable
    Exercises
    3.4 The Binomial Probability Distribution
    Exercises
    3.5 The Geometric Probability Distribution
    Exercises
    3.6 The Negative Binomial Probability Distribution (Optional)
    Exercises
    3.7 The Hypergeometric Probability Distribution
    Exercises
    3.8 The Poisson Probability Distribution
    Exercises
    3.9 Moments and Moment-Generating Functions
    Exercises
    3.10 Probability-Generating Functions (Optional)
    Exercises
    3.11 Tchebysheff’s Theorem
    Exercises
    3.12 Summary
    References and Further Readings
    Supplementary Exercises
    4 Continuous Variables and Their Probability Distributions
    4.1 Introduction
    4.2 The Probability Distribution for a Continuous Random Variable
    Exercises
    4.3 Expected Values for Continuous Random Variables
    Exercises
    4.4 The Uniform Probability Distribution
    Exercises
    4.5 The Normal Probability Distribution
    Exercises
    4.6 The Gamma Probability Distribution
    Exercises
    4.7 The Beta Probability Distribution
    Exercises
    4.8 Some General Comment
    4.9 Other Expected Values
    Exercises
    4.10 Tchebysheff’s Theorem
    Exercises
    4.11 Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)
    Exercises
    4.12 Summary
    References and Further Readings
    Supplementary Exercises
    5 Multivariate Probability Distributions
    5.1 Introduction
    5.2 Bivariate and Multivariate Probability Distributions
    Exercises
    5.3 Marginal and Conditional Probability Distributions
    Exercises
    5.4 Independent Random Variables
    Exercises
    5.5 The Expected Value of a Function of Random Variables
    5.6 Special Theorems
    Exercises
    5.7 The Covariance of Two Random Variables
    Exercises
    5.8 The Expected Value and Variance of Linear Functions of Random Variables
    Exercises
    5.9 The Multinomial Probability Distribution
    Exercises
    5.10 The Bivariate Normal Distribution (Optional)
    Exercises
    5.11 Conditional Expectations
    Exercises
    5.12 Summary
    References and Further Readings
    Supplementary Exercises
    6 Functions of Random Variables
    6.1 Introduction
    6.2 Finding the Probability Distribution of a Function of Random Variables
    6.3 The Method of Distribution Functions
    Exercises
    6.4 The Method of Transformations
    Exercises
    6.5 The Method of Moment-Generating Functions
    Exercises
    6.6 Multivariable Transformations Using Jacobians (Optional)
    Exercises
    6.7 Order Statistics
    Exercises
    6.8 Summary
    References and Further Readings
    Supplementary Exercises
    7 Sampling Distributions and the Central Limit Theorem
    7.1 Introduction
    Exercises
    7.2 Sampling Distributions Related to the Normal Distribution
    Exercises
    7.3 The Central Limit Theorem
    Exercises
    7.4 A Proof of the Central Limit Theorem (Optional)
    7.5 The Normal Approximation to the Binomial Distribution
    Exercises
    7.6 Summary
    References and Further Readings
    Supplementary Exercises
    8 Estimation
    8.1 Introduction
    8.2 The Bias and Mean Square Error of Point Estimators
    Exercises
    8.3 Some Common Unbiased Point Estimators
    8.4 Evaluating the Goodness of a Point Estimator
    Exercises
    8.5 Confidence Intervals
    Exercises
    8.6 Large-Sample Confidence Intervals
    Exercises
    8.7 Selecting the Sample Size
    Exercises
    8.8 Small-Sample Confidence Intervals for μ and μ[sub(1)] – μ[sub(2)]
    Exercises
    8.9 Confidence Intervals for σ[sup(2)]
    Exercises
    8.10 Summary
    References and Further Readings
    Supplementary Exercises
    9 Properties of Point Estimators and Methods of Estimation
    9.1 Introduction
    9.2 Relative Efficiency
    Exercises
    9.3 Consistency
    Exercises
    9.4 Sufficiency
    Exercises
    9.5 The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation
    Exercises
    9.6 The Method of Moments
    Exercises
    9.7 The Method of Maximum Likelihood
    Exercises
    9.8 Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional)
    Exercises
    9.9 Summary
    References and Further Readings
    Supplementary Exercises
    10 Hypothesis Testing
    10.1 Introduction
    10.2 Elements of a Statistical Test
    Exercises
    10.3 Common Large-Sample Tests
    Exercises
    10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests
    Exercises
    10.5 Relationships Between Hypothesis-Testing Procedures and Confidence Intervals
    Exercises
    10.6 Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values
    Exercises
    10.7 Some Comments on the Theory of Hypothesis Testing
    Exercises
    10.8 Small-Sample Hypothesis Testing for μ and μ[sub(1)] – μ[sub(2)]
    Exercises
    10.9 Testing Hypotheses Concerning Variances
    Exercises
    10.10 Power of Tests and the Neyman–Pearson Lemma
    Exercises
    10.11 Likelihood Ratio Tests
    Exercises
    10.12 Summary
    References and Further Readings
    Supplementary Exercises
    11 Linear Models and Estimation by Least Squares
    11.1 Introduction
    11.2 Linear Statistical Models
    11.3 The Method of Least Squares
    Exercises
    11.4 Properties of the Least-Squares Estimators: Simple Linear Regression
    Exercises
    11.5 Inferences Concerning the Parameters β[sub(i)]
    Exercises
    11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression
    Exercises
    11.7 Predicting a Particular Value of Y by Using Simple Linear Regression
    Exercises
    11.8 Correlation
    Exercises
    11.9 Some Practical Examples
    Exercises
    11.10 Fitting the Linear Model by Using Matrices
    Exercises
    11.11 Linear Functions of the Model Parameters: Multiple Linear Regression
    11.12 Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression
    Exercises
    11.13 Predicting a Particular Value of Y by Using Multiple Regression
    Exercises
    11.14 A Test for H[sub(0)]: β[sub(g+1)] = β[sub(g+2) = · · · = β[sub(k)] = 0
    Exercises
    11.15 Summary and Concluding Remarks
    References and Further Readings
    Supplementary Exercises
    12 Considerations in Designing Experiments
    12.1 The Elements Affecting the Information in a Sample
    12.2 Designing Experiments to Increase Accuracy
    Exercises
    12.3 The Matched-Pairs Experiment
    Exercises
    12.4 Some Elementary Experimental Designs
    Exercises
    12.5 Summary
    References and Further Readings
    Supplementary Exercises
    13 The Analysis of Variance
    13.1 Introduction
    13.2 The Analysis of Variance Procedure
    Exercises
    13.3 Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout
    13.4 An Analysis of Variance Table for a One-Way Layout
    Exercises
    13.5 A Statistical Model for the One-Way Layout
    Exercises
    13.6 Proof of Additivity of the Sums of Squares and E (MST) for a One-Way Layout (Optional)
    13.7 Estimation in the One-Way Layout
    Exercises
    13.8 A Statistical Model for the Randomized Block Design
    Exercises
    13.9 The Analysis of Variance for a Randomized Block Design
    Exercises
    13.10 Estimation in the Randomized Block Design
    Exercises
    13.11 Selecting the Sample Size
    Exercises
    13.12 Simultaneous Confidence Intervals for More Than One Parameter
    Exercises
    13.13 Analysis of Variance Using Linear Models
    Exercises
    13.14 Summary
    References and Further Readings
    Supplementary Exercises
    14 Analysis of Categorical Data
    14.1 A Description of the Experiment
    14.2 The Chi-Square Test
    14.3 A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test
    Exercises
    14.4 Contingency Tables
    Exercises
    14.5 r × c Tables with Fixed Row or Column Totals
    Exercises
    14.6 Other Applications
    14.7 Summary and Concluding Remarks
    References and Further Readings
    Supplementary Exercises
    15 Nonparametric Statistics
    15.1 Introduction
    15.2 A General Two-Sample Shift Model
    15.3 The Sign Test for a Matched-Pairs Experiment
    Exercises
    15.4 The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment
    Exercises
    15.5 Using Ranks for Comparing Two Population Distributions: Independent Random Samples
    15.6 The Mann–Whitney U Test: Independent Random Samples
    Exercises
    15.7 The Kruskal–Wallis Test for the One-Way Layout
    Exercises
    15.8 The Friedman Test for Randomized Block Designs
    Exercises
    15.9 The Runs Test: A Test for Randomness
    Exercises
    15.10 Rank Correlation Coefficient
    Exercises
    15.11 Some General Comments on Nonparametric Statistical Tests
    References and Further Readings
    Supplementary Exercises
    16 Introduction to Bayesian Methods for Inference
    16.1 Introduction
    16.2 Bayesian Priors, Posteriors, and Estimators
    Exercises
    16.3 Bayesian Credible Intervals
    Exercises
    16.4 Bayesian Tests of Hypotheses
    Exercises
    16.5 Summary and Additional Comments
    References and Further Readings
    Appendix 1 Matrices and Other Useful Mathematical Results
    A1.1 Matrices and Matrix Algebra
    A1.2 Addition of Matrices
    A1.3 Multiplication of a Matrix by a Real Number
    A1.4 Matrix Multiplication
    A1.5 Identity Elements
    A1.6 The Inverse of a Matrix
    A1.7 The Transpose of a Matrix
    A1.8 A Matrix Expression for a System of Simultaneous Linear Equations
    A1.9 Inverting a Matrix
    A1.10 Solving a System of Simultaneous Linear Equations
    A1.11 Other Useful Mathematical Results
    Appendix 2 Common Probability Distributions, Means, Variances, and Moment-Generating Functions
    Table 1 Discrete Distributions
    Table 2 Continuous Distributions
    Appendix 3 Tables
    Table 1 Binomial Probabilities
    Table 2 Table of e[sup(-x)]
    Table 3 Poisson Probabilities
    Table 4 Normal Curve Areas
    Table 5 Percentage Points of the t Distributions
    Table 6 Percentage Points of the χ2 Distributions
    Table 7 Percentage Points of the F Distributions
    Table 8 Distribution Function of U
    Table 9 Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50
    Table 10 Distribution of the Total Number of Runs R in Samples of Size (n1, n2); P(R < a)
    Table 11 Critical Values of Spearman’s Rank Correlation Coefficient
    Table 12 Random Numbers
    Answers to Exercises
    Index
    Continuous Distributions Distribution Probability Function Uniform f (y) = 2 θ 2 1 1 ; θ 1 − θ ≤ y ≤ θ exp −∞ < y < +∞ 1 2σ 2 − (y − µ)2 f (y) = 1 √ σ 2π Mean θ 1 + θ 2 2 Variance (θ 2 )2 1 − θ 12 Moment- Generating Function 2 − et θ − θ 1 ) et θ t (θ 2 1 µ β σ 2 β2 exp µt + t 2σ 2 2 (1 − βt )−1 f (y) = (α)β α −y/β ; αβ αβ2 (1 − βt )−α ; v 2v (1−2t)−v/2 f (y) = (α + β) (α)(β) yα−1(1 − y)β−1; 0 < y < 1 α α + β αβ (α + β)2(α + β + 1) does not exist in closed form Normal Exponential Gamma Chi-square Beta β f (y) = 1 β > 0 −y/β; e 0 < y < ∞ yα−1e 1 0 < y < ∞ −y/2 f (y) = (y)(v/2)−1e 2v/2(v/2) y2 > 0
    Discrete Distributions Distribution Binomial Geometric Hypergeometric Poisson Negative binomial Probability Function y = 0, 1, . . . , n n y py (1 − p)n−y; p(y) = p(y) = p(1 − p)y−1; y = 1, 2, . . . p(y) = ; y = 0, 1, . . . , n if n ≤ r, y = 0, 1, . . . , r if n > r N−r n−y N n r y −λ p(y) = λye ; y! y = 0, 1, 2, . . . y = r, r + 1, . . . y−1 r−1 p(y) = pr (1 − p)y−r ; Mean np Variance np(1 − p) Moment- Generating Function [ pet + (1 − p)]n 1 p nr N λ r p 1 − p p2 pet 1 − (1 − p)et n r N N − r N N − n N − 1 λ exp[λ(et − 1)] r(1 − p) p2 pet 1 − (1 − p)et r
    M A T H E M A T I C A L S T A T I S T I C S W I T H A P P L I C A T I O N S
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    S E V E N T H E D I T I O N Mathematical Statistics with Applications Dennis D. Wackerly University of Florida William Mendenhall III University of Florida, Emeritus Richard L. Scheaffer University of Florida, Emeritus Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States
    Mathematical Statistics with Applications, Seventh Edition Dennis D. Wackerly, William Mendenhall III, Richard L. Scheaffer Statistics Editor: Carolyn Crockett Assistant Editors: Beth Gershman, Catie Ronquillo Editorial Assistant: Ashley Summers Technology Project Manager: Jennifer Liang Marketing Manager: Mandy Jellerichs Marketing Assistant: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent Project Manager, Editorial Production: Hal Humphrey Art Director: Vernon Boes Print Buyer: Karen Hunt Production Service: Matrix Productions Inc. Copy Editor: Betty Duncan Cover Designer: Erik Adigard, Patricia McShane Cover Image: Erik Adigard Cover Printer: TK Compositor: International Typesetting and Composition Printer: TK © 2008, 2002 Duxbury, an imprint of Thomson Brooks/Cole, a part of The Thomson Corporation. Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license. Thomson Higher Education 10 Davis Drive Belmont, CA 94002-3098 USA For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com. Any additional questions about permissions can be submitted by e-mail to thomsonrights@thomson.com. ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher. Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 09 08 07 ExamView® and ExamView Pro® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. © 2008 Thomson Learning, Inc. All Rights Reserved. Thomson Learning WebTutorTM is a trademark of Thomson Learning, Inc. International Student Edition ISBN-13: 978-0-495-38508-0 ISBN-10: 0-495-38508-5
    CONTENTS Preface xiii Note to the Student xxi 1 What Is Statistics? 1 1.1 1.2 1.3 1.4 1.5 1.6 Introduction 1 Characterizing a Set of Measurements: Graphical Methods 3 Characterizing a Set of Measurements: Numerical Methods 8 How Inferences Are Made 13 Theory and Reality 14 Summary 15 2 Probability 20 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Introduction 20 Probability and Inference 21 A Review of Set Notation 23 A Probabilistic Model for an Experiment: The Discrete Case 26 Calculating the Probability of an Event: The Sample-Point Method 35 Tools for Counting Sample Points 40 Conditional Probability and the Independence of Events 51 Two Laws of Probability 57 v
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