Mathematical Statistics with Applications

Chapter 1: What is Statistics?

• • Introduction
• • Characterizing a Set of Measurements: Graphical Methods
• • Methods for Describing Sets of Data
• • Summary

Chapter 2: Probability

• • Introduction
• • Probability and Inference
• • A Review of Set Notation
• • A Probabilistic Model for an Experiment
• • 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 Method
• • The Law of Total Probability and Bayes' Rule
• • Numerical Events and Random Variables
• • Summary

Chapter 3: Discrete Random Variables and Their Probability Distributions

• • Basic Definition
• • The Probability Distribution for a Discrete Random Variable
• • Expected Values of Random Variables
• • The Binomial Probability Distribution
• • The Geometric Probability Distribution
• • The Hypergeometric Probability Distribution
• • The Poisson Probability Distribution
• • Moments and Moment-Generating Functions

Chapter 4: Continuous Random Variables and Their Probability Distributions

• • Introduction
• • The Probability Distribution for a Continuous Random Variable
• • Expected Values for Continuous Random Variables
• • The Uniform Probability Distribution
• • The Normal Probability Distribution
• • The Gamma Probability Distribution
• • The Beta Probability Distribution
• • Other Expected Values

Chapter 5: Multivariate Probability Distributions

• • Bivariate and Multivariate Probability Distributions
• • Marginal and Conditional Probability Distributions
• • Independent Random Variables
• • The Expected Value of a Function of Random Variables
• • Special Theorems
• • The Covariance of Two Random Variables
• • The Expected Value and Variance of Linear Functions of Random Variables
• • The Multinomial Probability Distribution

Chapter 6: Functions of Random Variables

• • Introduction
• • Finding the Probability Distribution of a Function of Random Variables
• • The Method of Distribution Functions
• • The Method of Transformations
• • The Method of Moment-Generating Functions
• • Order Statistics

Chapter 7: Sampling Distributions and the Central Limit Theorem

• • Introduction
• • Sampling Distributions Related to the Normal Distribution
• • The Central Limit Theorem
• • The Normal Approximation to the Binomial Distribution

Chapter 8: Estimation

• • Introduction
• • The Bias and Mean Square Error of Point Estimators
• • Common Point Estimators
• • Evaluating the Goodness of a Point Estimator
• • Confidence Intervals
• • Large-Sample Confidence Intervals
• • Selecting the Sample Size
• • Small-Sample Confidence Intervals for Mu and Mu1 - Mu2

Chapter 9: Properties of Point Estimators and Methods of Estimation

• • Introduction
• • Relative Efficiency
• • Consistency
• • Sufficiency
• • The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation
• • The Method of Moments
• • The Method of Maximum Likelihood

Chapter 10: Hypothesis Testing

• • Introduction
• • Elements of a Statistical Test
• • Common Large-Sample Tests
• • Calculating Type II Error Probabilities and Finding the Sample Size
• • Relationships Between Hypothesis-Testing Procedures and Confidence Intervals
• • Another Way to Report the Results of a Statistical Test: Attained Significance Levels or p-Values
• • Small-Sample Hypothesis Testing for Mu and Mu1 - Mu2
• • Power of Tests and the Neyman-Pearson Lemma

Chapter 11: Linear Models and Estimation by Least Squares

• • Introduction
• • Linear Statistical Models
• • The Method of Least Squares
• • Properties of the Least-Squares Estimators: Simple Linear Regression
• • Inferences Concerning the Parameters Beta_i
• • Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression

Chapter 12: Considerations in Designing Experiments

• • Introduction
• • The Elements of a Designing Experiment
• • The Randomized Block Design
• • The Latin Square Design

Chapter 13: The Analysis of Variance

• • Introduction
• • The Analysis of Variance Procedure
• • Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout
• • An Analysis of Variance Table for a One-Way Layout

Chapter 14: Nonparametric Statistics

• • Introduction
• • The Sign Test for a Matched Pairs Experiment
• • The Wilcoxon Signed-Rank Test
• • The Wilcoxon Rank Sum Test

Chapter 15: Bayesian Methods for Inference

• • Introduction
• • Bayesian Priors, Posteriors, and Estimators
• • Bayesian Credible Intervals
• • Bayesian Hypothesis Testing