Mathematical Statistics with Applications

by Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Overview

Intended for the two-term mathematical statistics course offered in mathematics and statistics departments (calculus prerequisite). This books presents a solid foundation in statistical theory, and at the same time, provides an indication of the relevance and importance of the theory in solving practical probelms in the real world.

Product Details

ISBN-13:	9780495110811
Publisher:	Cengage Learning
Publication date:	10/10/2007
Edition description:	New Edition
Pages:	944
Product dimensions:	9.46(w) x 7.76(h) x 1.51(d)

About the Author

Professor John T. Chen has been teaching mathematical statistics since 1998 at various universities including, The University of Sydney (1996-1997, Australia), McMaster University (1997-1998 Canada), University of Pittsburg (1998-2000), Bowling Green State University (2000-present), University of Michigan (2010 fall) and University of California, Berkeley (2017 fall). He has published two books, one on multivariate Bonferroni inequalities and another on prediction and statistical learning. Dr Chen’s research comprises of theoretical topics on probability inequalities, distribution theory and simultaneous inference. This aspect is featured by papers published in Biometrika, the Annals of the Institute of Mathematical Statistics, Journal of Applied Probability, among others. Besides theoretical statistics, his research also embraces applications of statistical methodologies to medical investigations and biostatistical consulting. This is reflected by papers published in Biometrics, the Annals of Neurology, The Annals of Thoracic Surgery, Journal of Vascular Surgery, among others. Dr. Chen enjoys cooperating rigorous research thinking and cutting-edge applications of statistical practices into classrooms to inspire students. With his experience and teaching efforts, Dr. Chen has earned teaching-related awards including, Teaching Excellence Awards by the Kappa Mu Epsilon Mathematics Honorary Society (2002 and 2006, BGSU chapter), Appreciations of Faculty Excellence (2019, 2020, 2021, BGSU), Certificate in Effective College Instruction recognized by the Association of College and University Educators and the American Council on Education (2023) and BGSU president’s Innovation award in AI teaching and learning (2024). Part of the materials in this book stem from his teaching notes and lesson plans accumulated over years of his enriched teaching experience.

Adam Loy is an associate professor of statistics at Carleton College. He teaches all levels of the statistics curriculum, including probability and mathematical statistics. Dr. Loy’s research focuses on incorporating realistic models, computation and visualization into the classroom, exploring the potential of visual inference, developing better visualizations to explore complex models and developing useful and usable R packages. He has publications in a variety of statistics journals including, the Journal of Statistics and Data Science Education, the Journal of Computational and Graphical Statistics and The R Journal, among others. Dr. Loy is currently an associate editor for both the Journal of Statistics and Data Science Education and the R Journal.

Table of Contents

1. WHAT IS STATISTICS? Characterizing a Set of Measurements: Graphical Methods. Characterizing a Set of Measurements: Numerical Methods. How Inferences Are Made. Theory and Reality. 2. PROBABILITY. Probability and Inference. A Review of Set Notation. A Probabilistic Model for an Experiment: The Discrete Case. Calculating the Probability of an Event: The Sample-Point Method. Tools for Use When Counting Sample Points. Conditional Probability and the Independence of Events. Two Laws of Probability. Calculating the Probability of an Event: The Event-Composition Method. The Law of Total Probability and Bayes's Rule. Numerical Events and Random Variables. Random Sampling. 3. DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS. Basic Definition. The Probability Distribution for a Discrete Random Variable. The Expected Value of a Random Variable or a Function of a Random Variable. The Binomial Probability Distribution. The Geometric Probability Distribution. The Negative Binomial Probability Distribution (Optional). The Hypergeometric Probability Distribution. The Poisson Probability Distribution. Moments and Moment-Generating Functions. Probability-Generating Functions (Optional). Tchebysheff's Theorem. 4. CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS. The Probability Distribution for a Continuous Random Variable. The Expected Value for a Continuous Random Variable. The Uniform Probability Distribution. The Normal Probability Distribution. The Gamma Probability Distribution. The Beta Probability Distribution. Some General Comments. Other Expected Values. Tchebysheff's Theorem. Expectations of Discontinuous Functions and Mixed ProbabilityDistributions (Optional). 5. MULTIVARIATE PROBABILITY DISTRIBUTIONS. Bivariate and Multivariate Probability Distributions. Marginal and Conditional Probability Distributions. Independent Random Variables. The Expected Value of a Function of Random Variables. Special Theorems. The Covariance of Two Random Variables. The Expected Value and Variance of Linear Functions of Random Variables. The Multinomial Probability Distribution. The Bivariate Normal Distribution (Optional). Conditional Expectations. 6. FUNCTIONS OF RANDOM VARIABLES. Finding the Probability Distribution of a Function of Random Variables. The Method of Distribution Functions. The Method of Transformations. The Method of Moment-Generating Functions. Order Statistics. 7. SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM. Sampling Distributions Related to the Normal Distribution. The Central Limit Theorem. A Proof of the Central Limit Theorem (Optional). The Normal Approximation to the Binomial Distribution. 8. ESTIMATION. The Bias and Mean Square Error of Point Estimators. Some Common Unbiased Point Estimators. Evaluating the Goodness of a Point Estimator. Confidence Intervals. Large-Sample Confidence Intervals. Selecting the Sample Size. Small-Sample Confidence Intervals for u (mu) and u1-u2. Confidence Intervals for s2. 9. PROPERTIES OF POINT ESTIMATORS AND METHODS OF ESTIMATION. Relative Efficiency. Consistency. Sufficiency. The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation. The Method of Moments. The Method of Maximum Likelihood. Some Large Sample Properties of MLEs (Optional). 10. HYPOTHESIS TESTING. Elements of a Statistical Test. Common Large-Sample Tests. Calculating Type II Error Probabilities and Finding the Sample Size for the Z Test. Relationships Between Hypothesis Testing Procedures and Confidence Intervals. Another Way to Report the Results of a Statistical Test: Attained Significance Levels or p-Values. Some Comments on the Theory of Hypothesis Testing. Small-Sample Hypothesis Testing for u (mu) and u1 - u2. Testing Hypotheses Concerning Variances. Power of Tests and the Neyman-Pearson Lemma. Likelihood Ratio Tests. 11. LINEAR MODELS AND ESTIMATION BY LEAST SQUARES. Linear Statistical Models. The Method of Least Squares. Fitting the Linear Model by Using Matrices. Properties of the Least Squares Estimators for the Model Y = B0 + B1x + E. Properties of the Least Squares Estimators for the Multiple Regression Model. Inferences Concerning the Parameters b1. Inferences Concerning Linear Functions of the Model Parameters. Predicting a Particular Value of Y. A Test to Test H1: Bg + I = ... = Bk = 0. Correlation. Some Practical Examples. 12. CONSIDERATIONS IN DESIGNING EXPERIMENTS. The Elements Affecting the Information in a Sample. The Physical Process of Designing an Experiment. Random Sampling and the Completely Randomized Design. Volume-Increasing Experimental Designs. Noise-Reducing Experimental Designs. 13. THE ANALYSIS OF VARIANCE. The Analysis of Variance Procedure. Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout. An Analysis of Variance Table for a One-Way Layout. A Statistical Model for the One-Way Layout. Proof of Additivity of the Sums of Squares and E(MST) for a One-Way Layout (Optional). Estimation in the One-Way Layout. A Statistical Model for the Randomized Block Design. The Analysis of Variance for a Randomized Block Design. Estimation in the Randomized Block Design. Selecting the Sample Size. Simultaneous Confidence Intervals for More Than One Parameter. Analysis of Variance Using Linear Models. 14. ANALYSIS OF CATEGORICAL DATA. A Description of the Experiment. The Chi-Square Test. A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test. Contingency Tables. r x c Tables with Fixed Row or Column Totals. Other Applications. 15. NONPARAMETRIC STATISTICS. A General Two-Sample Shift Model. The Sign Test for a Matched Pairs Experiment. The Wilcoxon Signed-Rank Test for a Matched Pairs Experiment. The Use of Ranks for Comparing Two Population Distributions: Independent Random Samples. The Mann-Whitney U Test: Independent Random Samples. The Kruskal-Wallis Test for the One-Way Layout. The Friedman Test for Randomized Block Designs. The Runs Test: A Test for Randomness. Rank Correlation Coefficient. Some General Comments on Nonparametric Statistical Tests. APPENDICES: 1. Matrices and Other Useful Mathematical Results. 2. Common Probability Distributions, Means, Variances, and Moment-Generating Functions. 3. Tables. ANSWERS TO EXERCISES. INDEX. (Each chapter begins with an Introduction and concludes with a Summary.)