Mathematical Statistics with Applications

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.

• 1. WHAT IS STATISTICS? Population and Data. Characterizing a Set of Measurements: Graphical Methods. Characterizing a Set of Measurements: Numerical Methods. Making Statistical Inference.2. PROBABILITY. Interpreting Probabilities. 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 Counting Sample Points. Conditional Probability and the Independence of Events. Two Laws of Probability. Calculating the Probability of an Event: The Event-Composition Methods. The Law of Total Probability and Bayes' Rule.3. DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS. Basic Definition. The Probability Distribution for Discrete Random Variable. The Expected Value of Random Variable or a Function of Random Variable. The Binomial Probability Distribution. The Geometric Probability Distribution. The Negative Binomial Probability Distribution (Optional). The Hypergeometric Probability Distribution. Moments and Moment-Generating Functions. Chebyshev's Inequality for Discrete Random Variables.4. CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS. The Probability Distribution for Continuous Random Variable. The Expected Value for Continuous Random Variable. The Uniform Probability Distribution. The Normal Probability Distribution. The Gamma Probability Distribution. The Beta Probability Distribution. Some General Comments. Moments and Moment-Generating Functions for Continuous Random Variables. Chebyshev's Inequality for Continuous Random Variables. Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional).5. MULTIVARIATE PROBABILITY DISTRIBUTIONS. Bivariate and Multivariate Probability Distributions. Independent Random Variables. The Expected Value of a Function of Random Variables. 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.Introductions. Finding the Probability Distribution of a Function of Random Variables. The Method of Distribution Functions. The Methods of Transformations. Multivariable Transformations Using Jacobians.7. SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM. Introduction. 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 Distributions. Order Statistics.8. ESTIMATION. The Bias and Mean Square Error of Point Estimators. Some Common Unbiased Point Estimators. Evaluating the Goodness of Point Estimator. Confidence Intervals. Large-Sample Confidence Intervals Selecting the Sample Size. Small-Sample Confidence Intervals for the Population Mean and Difference in Means. Confidence Intervals for the Population Variance.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: p-values. Some Comments on the Theory of Hypothesis Testing. Small-Sample Hypothesis Testing for the Population Mean and Difference in Means. Testing Hypotheses Concerning Variances. Power of Test and the Neyman-Pearson Lemma. Likelihood Ration Test.11. LINEAR MODELS AND ESTIMATION BY LEAST SQUARES. Linear Statistical Models. The Method of Least Squares. Properties of the Least Squares Estimators for the Simple Linear Regression Mode