Table of Contents

INTRODUCTION

REVIEW OF MATHEMATICS
Introduction
Combinatorics
Pascal's Triangle
Newton's Binomial Formula
Exponential Function
Stirling's Formula
Multinomial Theorem
Monotonic Functions
Convergence and Divergence
Taylor's Theorem
Differentiation and Summation
Some Properties of Integration
Integration by Parts
Region of Feasibility
Multiple Integration
Jacobian
Maxima and Minima
Lagrange Multiplier
L'Hôpital's Rule
Partial Fraction Expansion
Cauchy-Schwarz Inequality
Generating Functions
Difference Equations
Vectors, Matrices and Determinants
Real Numbers

PROBABILITY THEORY
Introduction
Subjective Probability, Relative Frequency and Empirical Probability
Sample Space
Decomposition of a Union of Events: Disjoint Events
Sigma Algebra and Probability Space
Rules and Axioms of Probability Theory
Conditional Probability
Law of Total Probability
Bayes Rule
Sampling With and Without Replacement
Probability and SIMULATION
Borel Sets
Measure Theory in Probability
Application of Probability Theory: Decision Analysis

RANDOM VARIABLES
Introduction
Discrete Random Variables
Cumulative Distribution Function
Continuous Random Variables
Joint Distributions
Independent Random Variables
Distribution of the Sum of Two Independent Random Variables
Moments, Expected Values and Variance
Covariance and Correlation
Distribution of a Function of a Random Variable
Multivariate Distributions and Marginal Densities
Conditional Expectations
Conditional Variance and Covariance
Moment Generating Functions
Characteristic Functions
Probability Generating Functions

DISCRETE DISTRIBUTIONS
Introduction
Bernoulli Distribution
Binomial Distribution
Multinomial Distribution
Hypergeometric Distribution
k-Variate Hypergeometric Distribution
Geometric Distribution
Negative Binomial Distribution
Negative Multinomial Distribution
Poisson Distribution
Discrete Uniform Distribution
Lesser Known Distributions
Joint Distributions
Convolutions
Compound Distributions
Branching Processes
Hierarchical Distributions

CONTINUOUS RANDOM VARIABLES
Location and Scale Parameters
Distribution of Functions of Random Variables
Uniform Distribution
Normal Distribution
Exponential Distribution
Poisson Process
Gamma Distribution
Beta Distribution
Chi-square Distribution
Student's t-Distribution
F-Distribution
Cauchy Distribution
Exponential Family
Hierarchical Models-Mixture Distributions
Other Distributions
Distributional Relationships
Additional Distributional Findings

DISTRIBUTIONS OF ORDER STATISTICS
Introduction
Rank Ordering
The Probability Integral Transformation
Distributions of Order Statistics in i.i.d. Samples
Expectations of Minimum and Maximum Order Statistics
Distributions of Single Order Statistics
Joint Distributions of Order Statistics

ASYMPTOTIC DISTRIBUTION THEORY
Introduction
Introducing Probability to the Limit Process
Introduction to Convergence in Distribution
Non-convergence
Introduction to Convergence in Probability
Convergence Almost Surely (with Probability One)
Convergence in rth Mean
Relationships Between Convergence Modalities
Application of Convergence in Distribution
Properties of Convergence in Probability
The Law of Large Numbers and Chebyshev's Inequality
The Central Limit Theorem
Proof of the Central Limit Theorem
The Delta Method
Convergence Almost Surely (with probability one)

POINT ESTIMATION
Introduction
Method of Moments Estimators
Maximum Likelihood Estimators
Bayes Estimators
Sufficient Statistics
Exponential Families
Other Estimators*
Criteria of a Good Point Estimator

HYPOTHESIS TESTING
Statistical Reasoning and Hypothesis Testing
Discovery, the Scientific Method, and Statistical Hypothesis Testing
Simple Hypothesis Testing
Statistical Significance
The Two Sample Test
Two Sided vs. One Sided Testing
Likelihood Ratios and the Neyman Pearson Lemma
One SampleTesting and the Normal Distribution
Two Sample Testing for the Normal Distribution
Likelihood Ratio Test and the Binomial Distribution
Likelihood Ratio Test and the Poisson Distribution
The Multiple Testing Issue
Nonparametric Testing
Goodness of Fit Testing
Fisher's Exact Test
Sample Size Computations

INTERVAL ESTIMATION
Introduction
Definition
Constructing Confidence Intervals
Bayesian Credible Intervals
Approximate Confidence Intervals and MLE Pivot
The Bootstrap Method*
Criteria of a Good Interval Estimator
Confidence Intervals and Hypothesis Tests

INTRODUCTION TO COMPUTATIONAL METHODS
The Newton-Raphson Method
The EM Algorithm
Simulation
Markov Chains
Markov Chain Monte Carlo Methods

INDEX