Introduction to Probability and Statistics

Vijay K. Rohatgi, PhD, is Professor Emeritus in the Department of Mathematics and Statistics at Bowling Green State University.  An Investment Research Consultant for PRI Investments, he is also the author of several books and over 100 research papers. A. K. Md. Ehsanes Saleh, PhD, is Distinguished Research Professor in the School of Mathematics and Statistics at Carleton University. Dr. Saleh is the author of more than 200 journal articles, and his research interests include nonparametric statistics, order statistics, and robust estimation.

"The book is an ideal reference and resource for scientists and engineers in the elds of statistics, mathematics, physics, industrial management, and engineering. The book is also an excellent text for upper-undergraduate and graduate-level students majoring in probability and statistics." (Zentralblatt MATH, 2016)

• PREFACE TO THE THIRD EDITION xiiiPREFACE TO THE SECOND EDITION xvPREFACE TO THE FIRST EDITION xviiACKNOWLEDGMENTS xixENUMERATION OF THEOREMS AND REFERENCES xxi1 Probability 11.1 Introduction 11.2 Sample Space 21.3 Probability Axioms 71.4 Combinatorics: Probability on Finite Sample Spaces 201.5 Conditional Probability and Bayes Theorem 261.6 Independence of Events 312 Random Variables and Their Probability Distributions 392.1 Introduction 392.2 Random Variables 392.3 Probability Distribution of a Random Variable 422.4 Discrete and Continuous Random Variables 472.5 Functions of a Random Variable 553 Moments and Generating Functions 673.1 Introduction 673.2 Moments of a Distribution Function 673.3 Generating Functions 833.4 Some Moment Inequalities 934 Multiple Random Variables 994.1 Introduction 994.2 Multiple Random Variables 994.3 Independent Random Variables 1144.4 Functions of Several Random Variables 1234.5 Covariance, Correlation and Moments 1434.6 Conditional Expectation 1574.7 Order Statistics and Their Distributions 1645 Some Special Distributions 1735.1 Introduction 1735.2 Some Discrete Distributions 1735.2.1 Degenerate Distribution 1735.2.2 Two-Point Distribution 1745.2.3 Uniform Distribution on n Points 1755.2.4 Binomial Distribution 1765.2.5 Negative Binomial Distribution (Pascal or Waiting Time Distribution) 1785.2.6 Hypergeometric Distribution 1835.2.7 Negative Hypergeometric Distribution 1855.2.8 Poisson Distribution 1865.2.9 Multinomial Distribution 1895.2.10 Multivariate Hypergeometric Distribution 1925.2.11 Multivariate Negative Binomial Distribution 1925.3 Some Continuous Distributions 1965.3.1 Uniform Distribution (Rectangular Distribution) 1995.3.2 Gamma Distribution 2025.3.3 Beta Distribution 2105.3.4 Cauchy Distribution 2135.3.5 Normal Distribution (the Gaussian Law) 2165.3.6 Some Other Continuous Distributions 2225.4 Bivariate and Multivariate Normal Distributions 2285.5 Exponential Family of Distributions 2406 Sample Statistics and Their Distributions 2456.1 Introduction 2456.2 Random Sampling 2466.3 Sample Characteristics and Their Distributions 2496.4 Chi-Square, t-, and F-Distributions: Exact Sampling Distributions 2626.5 Distribution of (X,S2) in Sampling from a Normal Population 2716.6 Sampling from a Bivariate Normal Distribution 2767 Basic Asymptotics: Large Sample Theory 2857.1 Introduction 2857.2 Modes of Convergence 2857.3 Weak Law of Large Numbers 3027.4 Strong Law of Large Numbers 3087.5 Limiting Moment Generating Functions 3167.6 Central Limit Theorem 3217.7 Large Sample Theory 3318 Parametric Point Estimation 3378.1 Introduction 3378.2 Problem of Point Estimation 3388.3 Sufficiency, Completeness and Ancillarity 3428.4 Unbiased Estimation 3598.5 Unbiased Estimation (Continued): A Lower Bound for the Variance of An Estimator 3728.6 Substitution Principle (Method of Moments) 3868.7 Maximum Likelihood Estimators 3888.8 Bayes and Minimax Estimation 4018.9 Principle of Equivariance 4189 Neyman–Pearson Theory of Testing of Hypotheses 4299.1 Introduction 4299.2 Some Fundamental Notions of Hypotheses Testing 4299.3 Neyman–Pearson Lemma 4389.4 Families with Monotone Likelihood Ratio 4469.5 Unbiased and Invariant Tests 4539.6 Locally Most Powerful Tests 45910 Some Further Results on Hypotheses Testing 46310.1 Introduction 46310.2 Generalized Likelihood Ratio Tests 46310.3 Chi-Square Tests 47210.4 t-Tests 48410.5 F-Tests 48910.6 Bayes and Minimax Procedures 49111 Confidence Estimation 49911.1 Introduction 49911.2 Some Fundamental Notions of Confidence Estimation 49911.3 Methods of Finding Confidence Intervals 50411.4 Shortest-Length Confidence Intervals 51711.5 Unbiased and Equivariant Confidence Intervals 52311.6 Resampling: Bootstrap Method 53012 General Linear Hypothesis 53512.1 Introduction 53512.2 General Linear Hypothesis 53512.3 Regression Analysis 54312.3.1 Multiple Linear Regression 54312.3.2 Logistic and Poisson Regression 55112.4 One-Way Analysis of Variance 55412.5 Two-Way Analysis of Variance with One Observation Per Cell 56012.6 Two-Way Analysis of Variance with Interaction 56613 Nonparametric Statistical Inference 57513.1 Introduction 57513.2 U-Statistics 57613.3 Some Single-Sample Problems 58413.3.1 Goodness-of-Fit Problem 58413.3.2 Problem of Location 59013.4 Some Two-Sample Problems 59913.4.1 Median Test 60113.4.2 Kolmogorov–Smirnov Test 60213.4.3 The Mann–Whitney–Wilcoxon Test 60413.5 Tests of Independence 60813.5.1 Chi-square Test of Independence—Contingency Tables 60813.5.2 Kendall’s Tau 61113.5.3 Spearman’s Rank Correlation Coefficient 61413.6 Some Applications of Order Statistics 61913.7 Robustness 62513.7.1 Effect of Deviations from Model Assumptions on Some Parametric Procedures 62513.7.2 Some Robust Procedures 631FREQUENTLY USED SYMBOLS AND ABBREVIATIONS 637REFERENCES 641STATISTICAL TABLES 647ANSWERS TO SELECTED PROBLEMS 667AUTHOR INDEX 677SUBJECT INDEX 679