An Introduction to Probability and Mathematical Statistics

PrefaceChapter 1 Events and Probabilities 1.1 Combinatorial Probability 1.2 The Fundamental Probability Set and the Algebra of Events 1.3 The Axioms of a Probability SpaceChapter 2 Dependent and Independent Events 2.1 Conditional Probability 2.2 Stochastic Independence 2.3 An Application in Physics of the Notion of IndependenceChapter 3 Random Variables and Probability Distributions 3.1 The Definition of a Function 3.2 The Definition of a Random Variable 3.3 Combinations of Random Variables 3.4 Distribution Functions 3.5 Multivariate Distribution FunctionsChapter 4 Discrete Distributions 4.1 Univariate Discrete Distributions 4.2 The Binomial and Pascal Distributions 4.3 The Hypergeometric Distribution 4.4 The Poisson Distribution 4.5 Multivariate Discrete DensitiesChapter 5 Absolutely Continuous Distributions 5.1 Absolutely Continuous Distributions 5.2 Densities of Functions of Random VariablesChapter 6 Some Special Absolutely Continuous Distributions 6.1 The Gamma and Beta Functions 6.2 The Normal Distribution 6.3 The Negative Exponential Distribution 6.4 The Chi-Square Distribution 6.5 The F-Distribution and the t-DistributionChapter 7 Expectation and Limit Theorems 7.1 Definition of Expectation 7.2 Expectation of Functions of Random Variables 7.3 Moments and Central Moments 7.4 Convergence in Probability 7.5 Limit TheoremsChapter 8 Point Estimation 8.1 Sampling 8.2 Unbiased and Consistent Estimates 8.3 The Method of Moments 8.4 Minimum Variance Estimates 8.5 The Principle of Maximum LikelihoodChapter 9 Notes on Matrix TheoryChapter 10 The Multivariate Normal Distribution 10.1 The Multivariate Normal Density 10.2 Properties of the Multivariate Normal Distribution 10.3 Cochran's Theorem 10.4 Proof of the Independence of the Sample Mean and Sample Variance for a Normal PopulationChapter 11 Testing Statistical Hypotheses: Simple Hypothesis vs. Simple Alternative 11.1 Fundamental Notions of Hypothesis Testing 11.2 Simple Hypothesis vs. Simple Alternative 11.3 The Neyman-Pearson Fundamental Lemma 11.4 Randomized TestsChapter 12 Testing Simple and Composite Hypotheses 12.1 Uniformly Most Powerful Critical Regions 12.2 The Likelihood Ratio Test 12.3 The t-Test 12.4 The Analysis of VarianceChapter 13 Confidence Intervals 13.1 The Neyman Theory of Confidence Intervals 13.2 The Relation Between Confidence Intervals and Tests of Hypotheses 13.3 Necessary and Sufficient Conditions for the Existence of Confidence IntervalsSuggested ReadingTables I—IVIndex