AN INTRODUCTION TO PROBABILITY AND STATISTICAL INFERENCE

By N. NOANI

CONTENTS SOME MOTIVATING EXAMPLES AND SOME FUNDAMENTAL CONCEPTS SOME MOTIVATING EXAMPLES SOME FUNDAMENTAL CONCEPTS RANDOM VARIABLES THE CONCEPT OF PROBABILITY AND BASIC RESULTS DE?NITION OF PROBABILITY AND SOME BASIC RESULTS DISTRIBUTION OF A RANDOM VARIABLE CONDITIONAL PROBABILITY AND RELATED RESULTS INDEPENDENT EVENTS AND RELATED RESULTS BASIC CONCEPTS AND RESULTS IN COUNTING NUMERICAL CHARACTERISTICS OF A RANDOM VARIABLE, SOME SPECIAL RANDOM VARIABLES EXPECTATION, VARIANCE, AND MOMENT GENERATING FUNCTION OF A RANDOM VARIABLE SOME PROBABILITY INEQUALITIES SOME SPECIAL RANDOM VARIABLES MEDIAN AND MODE OF A RANDOM VARIABLE JOINT AND CONDITIONAL P D F ’S, CONDITIONAL EXPECTATION AND VARIANCE, MOMENT GENERATING FUNCTION, COVARIANCE, AND CORRELATION COEFFICIENT JOINT D F AND JOINT P D F OF TWO RANDOM VARIABLES MARGINAL AND CONDITIONAL P D F ’S, CONDITIONAL EXPECTATION AND VARIANCE EXPECTATION OF A FUNCTION OF TWO R V ’S, JOINT AND MARGINAL M G F ’S, COVARIANCE, AND CORRELATION COEF?CIENT SOME GENERALIZATIONS TO K RANDOM VARIABLES THE MULTINOMIAL, THE BIVARIATE NORMAL, AND THE MULTIVARIATE NORMAL DISTRIBUTIONS INDEPENDENCE OF RANDOM VARIABLES AND SOME APPLICATIONS INDEPENDENCE OF RANDOM VARIABLES AND CRITERIA OF INDEPENDENCE THE REPRODUCTIVE PROPERTY OF CERTAIN DISTRIBUTIONS TRANSFORMATION OF RANDOM VARIABLES TRANSFORMING A SINGLE RANDOM VARIABLE TRANSFORMING TWO OR MORE RANDOM VARIABLES LINEAR TRANSFORMATIONS THE PROBABILITY INTEGRAL TRANSFORM ORDER STATISTICS SOME MODES OF CONVERGENCE OF RANDOM VARIABLES, APPLICATIONS CONVERGENCE IN DISTRIBUTION OR IN PROBABILITY AND THEIR RELATIONSHIP SOME APPLICATIONS OF CONVERGENCE IN DISTRIBUTION: THE WEAK LAW OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM FURTHER LIMIT THEOREMS AN OVERVIEW OF STATISTICAL INFERENCE THE BASICS OF POINT ESTIMATION THE BASICS OF INTERVAL ESTIMATION THE BASICS OF TESTING HYPOTHESES THE BASICS OF REGRESSION ANALYSIS THE BASICS OF ANALYSIS OF VARIANCE THE BASICS OF NONPARAMETRIC INFERENCE POINT ESTIMATION MAXIMUM LIKELIHOOD ESTIMATION: MOTIVATION AND EXAMPLES SOME PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATES UNIFORMLY MINIMUM VARIANCE UNBIASED ESTIMATES DECISION-THEORETIC APPROACH TO ESTIMATION OTHER METHODS OF ESTIMATION CONFIDENCE INTERVALS AND CONFIDENCE REGIONS CON?DENCE INTERVALS CON?DENCE INTERVALS IN THE PRESENCE OF NUISANCE PARAMETERS A CON?DENCE REGION FOR (?, ?2) IN THE N(?, ?2) DISTRIBUTION CON?DENCE INTERVALS WITH APPROXIMATE CON?DENCE COEF?CIENT TESTING HYPOTHESES GENERAL CONCEPTS, FORMULATION OF SOME TESTING HYPOTHESES NEYMAN–PEARSON FUNDAMENTAL LEMMA, EXPONENTIAL TYPE FAMILIES, UNIFORMLY MOST POWERFUL TESTS FOR SOME COMPOSITE HYPOTHESES SOME APPLICATIONS OF THEOREMS 2 AND LIKELIHOOD RATIO TESTS MORE ABOUT TESTING HYPOTHESES LIKELIHOOD RATIO TESTS IN THE MULTINOMIAL CASE AND CONTINGENCY TABLES A GOODNESS-OF-FIT TEST DECISION-THEORETIC APPROACH TO TESTING HYPOTHESES RELATIONSHIP BETWEEN TESTING HYPOTHESES AND CON?DENCE REGIONS A SIMPLE LINEAR REGRESSION MODEL SETTING-UP THE MODEL — THE PRINCIPLE OF LEAST SQUARES THE LEAST SQUARES ESTIMATES OF ?1 AND ?2, AND SOME OF THEIR PROPERTIES NORMALLY DISTRIBUTED ERRORS: MLE’S OF ?1, ?2, AND ?2, SOME DISTRIBUTIONAL RESULTS CON?DENCE INTERVALS AND HYPOTHESES TESTING PROBLEMS SOME PREDICTION PROBLEMS PROOF OF THEOREM CONCLUDING REMARKS TWO MODELS OF ANALYSIS OF VARIANCE ONE-WAY LAYOUT WITH THE SAME NUMBER OF OBSERVATIONS PER CELL A MULTICOMPARISON METHOD TWO-WAY LAYOUT WITH ONE OBSERVATION PER CELL SOME TOPICS IN NONPARAMETRIC INFERENCE SOME CON?DENCE INTERVALS WITH GIVEN APPROXIMATE CON?DENCE COEF?CIENT CON?DENCE INTERVALS FOR QUANTILES OF A DISTRIBUTION FUNCTION THE TWO-SAMPLE SIGN TEST THE RANK SUM AND THE WILCOXON–MANN–WHITNEY TWO-SAMPLE TESTS NONPARAMETRIC CURVE ESTIMATION