Probability and Statistical Inference
OverviewBOOK DESCRIPTION: Written by two leading statisticians, this applied introduction to the mathematics of probability and statistics emphasizes the existence of variation in almost every process, and how the study of probability and statistics helps us understand this variation. Designed for students with a background in calculus, this book continues to reinforce basic mathematical concepts with numerous real-world examples and applications to illustrate the relevance of key concepts. NEW TO THIS EDITION: The included CD-ROM contains all of the data sets in a variety of formats for use with most statistical software packages. This disc also includes several applications of Minitab(R) and Maple(TM). Historical vignettes at the end of each chapter outline the origin of the greatest accomplishments in the field of statistics, adding enrichment to the course. Content updatesThe first five chapters have been reorganized to cover a standard probability course with more real examples and exercises. These chapters are important for students wishing to pass the first actuarial exam, and cover the necessary material needed for students taking this course at the junior level. Chapters 6 and 7 on estimation and tests of statistical hypotheses tie together confidence intervals and tests, including one-sided ones. There are separate chapters on nonparametric methods, Bayesian methods, and Quality Improvement. Chapters 4 and 5 include a strong discussion on conditional distributions and functions of random variables, including Jacobians of transformations and the moment-generating technique. Approximations of distributions like the binomial and the Poisson with the normal can be found using the central limit theorem. Chapter 8 (Nonparametric Methods) includes most of the standards tests such as those by Wilcoxon and also the use of order statistics in some distribution-free inferences. Chapter 9 (Bayesian Methods) explains the use of the "Dutch book" to prove certain probability theorems. Chapter 11 (Quality Improvement) stresses how important W. Edwards Deming's ideas are in understanding variation and how they apply to everyday life. TABLE OF CONTENTS: PrefacePrologue 1. Probability1.1 Basic Concepts1.2 Properties of Probability1.3 Methods of Enumeration1.4 Conditional Probability1.5 Independent Events1.6 Bayes's Theorem 2. Discrete Distributions2.1 Random Variables of the Discrete Type2.2 Mathematical Expectation2.3 The Mean, Variance, and Standard Deviation2.4 Bernoulli Trials and the Binomial Distribution2.5 The Moment-Generating Function2.6 The Poisson Distribution 3. Continuous Distributions3.1 Continuous-Type Data3.2 Exploratory Data Analysis3.3 Random Variables of the Continuous Type3.4 The Uniform and Exponential Distributions3.5 The Gamma and Chi-Square Distributions3.6 The Normal Distribution3.7 Additional Models 4. Bivariate Distributions4.1 Distributions of Two Random Variables4.2 The Correlation Coefficient4.3 Conditional Distributions4.4 The Bivariate Normal Distribution 5. Distributions of Functions of Random Variables5.1 Functions of One Random Variable5.2 Transformations of Two Random Variables5.3 Several Independent Random Variables5.4 The Moment-Generating Function Technique5.5 Random Functions Associated with Normal Distributions5.6 The Central Limit Theorem5.7 Approximations for Discrete Distributions 6. Estimation6.1 Point Estimation6.2 Confidence Intervals for Means6.3 Confidence Intervals for Difference of Two Means6.4 Confidence Intervals for Variances6.5 Confidence Intervals for Proportions6.6 Sample Size.6.7 A Simple Regression Problem6.8 More Regression 7. Tests of Statistical Hypotheses7.1 Tests about Proportions7.2 Tests about One Mean7.3 Tests of the Equality of Two Means7.4 Tests for Variances7.5 One-Factor Analysis of Variance7.6 Two-Factor Analysis of Variance7.7 Tests Concerning Regression and Correlation 8. Nonparametric Methods8.1 Chi-Square Goodness of Fit Tests8.2 Contingency Tables8.3 Order Statistics8.4 Distribution-Free Confidence Intervals for Percentiles8.5 The Wilcoxon Tests8.6 Run Test and Test for Randomness8.7 Kolmogorov-Smirnov Goodness of Fit Test8.8 Resampling Methods 9. Bayesian Methods9.1 Subjective Probability9.2 Bayesian Estimation9.3 More Bayesian Concepts 10. Some Theory10.1 Sufficient Statistics10.2 Power of a Statistical Test10.3 Best Critical Regions10.4 Likelihood Ratio Tests10.5 Chebyshev's Inequality and Convergence in Probability10.6 Limiting Moment-Generating Functions10.7 Asymptotic Distributions of Maximum Likelihood Estimators 11. Quality Improvement Through Statistical Methods11.1 Time Sequences11.2 Statistical Quality Control11.3 General Factorial and 2k Factorial Designs11.4 Understanding Variation A. Review of Selected Mathematical Techniques A.1 Algebra of Sets A.2 Mathematical Tools for the Hypergeometric Distribution A.3 Limits A.4 Infinite Series A.5 Integration A.6 Multivariate CalculusB. ReferencesC. TablesD. Answers to Odd-Numbered Exercises
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