Probability and Statistical Inference

PROBABILITY AND STATISTICAL INFERENCE
	CONTENTS
	Preface
	1 Experiments, Sample Spaces, and Events
		1.1 Introduction
		1.2 Sample Space
		1.3 Algebra of Events
		1.4 Infinite Operations on Events
	2 Probability
		2.1 Introduction
		2.2 Probability as a Frequency
		2.3 Axioms of Probability
		2.4 Consequences of the Axioms
		2.5 Classical Probability
		2.6 Necessity of the Axioms
		2.7 Subjective Probability
	3 Counting
		3.1 Introduction
		3.2 Product Sets, Orderings, and Permutations
		3.3 Binomial Coefficients
		3.4 Extension of Newton’s Formula
		3.5 Multinomial Coefficients
	4 Conditional Probability; Independence
		4.1 Introduction
		4.2 Conditional Probability
		4.3 Partitions; Total Probability Formula
		4.4 Bayes’ Formula
		4.5 Independence
		4.6 Exchangeability; Conditional Independence
	5 Markov Chains
		5.1 Introduction and Basic Definitions
		5.2 Definition of a Markov Chain
		5.3 n-Step Transition Probabilities
		5.4 The Ergodic Theorem
		5.5 Absorption Probabilities
	6 Random Variables: Univariate Case
		6.1 Introduction
		6.2 Distributions of Random Variables
		6.3 Discrete and Continuous Random Variables
		6.4 Functions of Random Variables
		6.5 Survival and Hazard Functions
	7 Random Variables: Multivariate Case
		7.1 Bivariate Distributions
		7.2 Marginal Distributions; Independence
		7.3 Conditional Distributions
		7.4 Bivariate Transformations
		7.5 Multidimensional Distributions
	8 Expectation
		8.1 Introduction
		8.2 Expected Value
		8.3 Expectation as an Integral
		8.4 Properties of Expectation
		8.5 Moments
		8.6 Variance
		8.7 Conditional Expectation
		8.8 Inequalities
	9 Selected Families of Distributions
		9.1 Bernoulli Trials and Related Distributions
		9.2 Hypergeometric Distribution
		9.3 Poisson Distribution and Poisson Process
		9.4 Exponential, Gamma and Related Distributions
		9.5 Normal Distribution
		9.6 Beta Distribution
	10 Random Samples
		10.1 Statistics and their Distributions
		10.2 Distributions Related to Normal
		10.3 Order Statistics
		10.4 Generating Random Samples
		10.5 Convergence
		10.6 Central Limit Theorem
	11 Introduction to Statistical Inference
		11.1 Overview
		11.2 Descriptive Statistics
		11.3 Basic Model
		11.4 Bayesian Statistics
		11.5 Sampling
		11.6 Measurement Scales
	12 Estimation
		12.1 Introduction
		12.2 Consistency
		12.3 Loss, Risk, and Admissibility
		12.4 Efficiency
		12.5 Methods of Obtaining Estimators
		12.6 Sufficiency
		12.7 Interval Estimation
	13 Testing Statistical Hypotheses
		13.1 Introduction
		13.2 Intuitive Background
		13.3 Most Powerful Tests
		13.4 Uniformly Most Powerful Tests
		13.5 Unbiased Tests
		13.6 Generalized Likelihood Ratio Tests
		13.7 Conditional Tests
		13.8 Tests and Confidence Intervals
		13.9 Review of Tests for Normal Distributions
		13.10 Monte Carlo, Bootstrap, and Permutation Tests
	14 Linear Models
		14.1 Introduction
		14.2 Regression of the First and Second Kind
		14.3 Distributional Assumptions
		14.4 Linear Regression in the Normal Case
		14.5 Testing Linearity
		14.6 Prediction
		14.7 Inverse Regression
		14.8 BLUE
		14.9 Regression Toward the Mean
		14.10 Analysis of Variance
		14.11 One-way Layout
		14.12 Two-way Layout
		14.13 ANOVA Models with Interaction
		14.14 Further Extensions
	15 Rank Methods
		15.1 Introduction
		15.2 Glivenko-Cantelli Theorem
		15.3 Kolmogorov-Smirnov Tests
		15.4 One-Sample Rank Tests
		15.5 Two-Sample Rank Tests
		15.6 Kruskal-Wallis Test
	16 Analysis of Categorical Data
		16.1 Introduction
		16.2 Chi-square Tests
		16.3 Homogeneity and Independence
		16.4 Consistency and Power
		16.5 2×2 Contingency Tables
		16.6 r × c Contingency Tables
	Statistical Tables
	Bibliography
	Answers to Odd-Numbered Problems
	Index