Probability and Statistics

1 Probability Models
1.1 Probability: A Measure of Uncertainty
1.2 Probability Models
1.3 Properties of Probability Models
1.4 Uniform Probability on Finite Spaces
1.5 Conditional Probability and Independence
1.6 Continuity of P
1.7 Further Proofs (Advanced)

2 Random Variables and Distributions
2.1 Random Variables
2.2 Distributions of Random Variables
2.3 Discrete Distributions
2.4 Continuous Distributions
2.5 Cumulative Distribution Functions
2.6 One-Dimensional Change of Variable
2.7 Joint Distributions
2.8 Conditioning and Independence
2.9 Multidimensional Change of Variable
2.10 Simulating Probability Distributions

3 Expectation
3.1 The Discrete Case
3.2 The Absolutely Continuous Case
3.3 Variance, Covariance, and Correlation
3.4 Generating Functions
3.5 Conditional Expectation
3.6 Inequalities
3.7 General Expectations (Advanced)
3.8 Further Proofs (Advanced)

4 Sampling Distributions and Limits
4.1 Sampling Distributions
4.2 Convergence in Probability
4.3 Convergence with Probability 1
4.4 Convergence in Distribution
4.5 Monte Carlo Approximations
4.6 Normal Distribution Theory
4.7 Further Proofs (Advanced)

5 Statistical Inference
5.1 Why Do We Need Statistics?
5.2 Inference Using a Probability Model
5.3 Statistical Models
5.4 Data Collection
5.5 Some Basic Inferences

6 Likelihood Inference
6.1 The Likelihood Function
6.2 Maximum Likelihood Estimation
6.3 Inferences Based on the MLE
6.4 Distribution-Free Methods
6.5 Asymptotics for the MLE (Advanced)

7 Bayesian Inference
7.1 The Prior and Posterior Distributions
7.2 Inference Based on the Posterior
7.3 Bayesian Computations
7.4 Choosing Priors
7.5 Further Proofs (Advanced)

8 Optimal Inferences
8.1 Optimal Unbiased Estimation
8.2 Optimal Hypothesis Testing
8.3 Optimal Bayesian Inferences
8.4 Decision Theory (Advanced)
8.5 Further Proofs (Advanced)

9 Model Checking
9.1 Checking the Sampling Model
9.2 Checking for Prior-Data Conflict
9.3 The Problem with Multiple Checks

10 Relationships Among Variables
10.1 Related Variables
10.2 Categorical Response and Predictors
10.3 Quantitative Response and Predictors
10.4 Quantitative Response and Categorical Predictors
10.5 Categorical Response and Quantitative Predictors
10.6 Further Proofs (Advanced)

11 Advanced Topic -Stochastic Processes
11.1 Simple Random Walk
11.2 Markov Chains
11.3 Markov Chain Monte Carlo
11.4 Martingales
11.5 Brownian Motion
11.6 Poisson Processes
11.7 Further Proofs

Appendices
A Mathematical Background
A.1 Derivatives
A.2 Integrals
A.3 Infinite Series
A.4 Matrix Multiplication
A.5 Partial Derivatives
A.6 Multivariable Integrals

B Computations
B.1 Using R
B.2 Using Minitab

C Common Distributions
C.1 Discrete Distributions
C.2 Absolutely Continuous Distributions

D Tables
D.1 Random Numbers
D.2 Standard Normal Cdf
D.3 Chi-Squared Distribution Quantiles
D.4 t Distribution Quantiles
D.5 F Distribution Quantiles
D.6 Binomial Distribution Probabilities

E Answers to Odd-Numbered Exercises
Index