Proabability and Statistics: The Science of Uncert

Unlike traditional introductory math/stat textbooks, Probability and Statistics: The Science of Uncertainty brings a modern flavor based on incorporating the computer to the course and an integrated approach to inference. From the start the book integrates simulations into its theoretical coverage, and emphasizes the use of computer-powered computation throughout.* Math and science majors with just one year of Calculus can use this text and experience a refreshing blend of applications and theory that goes beyond merely mastering the technicalities. They'll get a thorough grounding in probability theory, and go beyond that to the theory of statistical inference and its applications. An integrated approach to inference is presented that includes the frequency approach as well as Bayesian methodology. Bayesian inference is developed as a logical extension of likelihood methods. A separate chapter is devoted to the important topic of model checking and this is applied in the context of the standard applied statistical techniques. Examples of data analyses using real-world data are presented throughout the text. A final chapter introduces a number of the most important stochastic process models using elementary methods.

Contents

1. Probability Models
    1.1 Probability: A Measure of Uncertainty
       1.1.1 Why Do We Need Probability Theory?
    1.2 Probability Models
    1.3 Basic Results for Probability Models
    1.4 Uniform Probability on Finite Spaces
       1.4.1 Combinatorial Principles
    1.5 Conditional Probability and Independence
       1.5.1 Conditional Probability
       1.5.2 Independence of Events
    1.6 Continuity of P
    1.7 Further Proofs (Advanced)
    
  2. Random Variables and Distributions
    2.1 Random Variables
    2.2 Distribution of Random Variables
    2.3 Discrete Distributions
       2.3.1 Important Discrete Distributions
    2.4 Continuous Distributions
       2.4.1 Important Absolutely Continuous Distributions
    2.5 Cumulative Distribution Functions (cdfs)
       2.5.1 Properties of Distribution Functions
       2.5.2 Cdf's of Discrete Distributions
       2.5.3 Cdf's of Absolutely Continuous Distributions
       2.5.4 Mixture Distributions
       2.5.5 Distributions Neither Discrete Nor Continuous (Advanced)
    2.6 One-dimensional Change of Variable
       2.6.1 The Discrete Case
       2.6.2 The Continuous Case
    2.7 Joint Distributions
       2.7.1 Joint Cumulative Distribution Functions
       2.7.2 Marginal Distributions
       2.7.3 Joint Probability Functions
       2.7.4 Joint Density Functions
    2.8 Conditioning and Independence
       2.8.1 Conditioning on Discrete Random Variables
       2.8.2 Conditioning on Continuous Random Variables
       2.8.3 Independence of Random Variables
       2.8.4 Sampling From a Population
    2.9 Multi-dimensional Change of Variable
       2.9.1 The Discrete Case
       2.9.2 The Continuous Case (Advance)
       2.9.3 Convolution
    2.10 Simulating Probability Distributions
       2.10.1 Simulating Discrete Distributions
       2.10.2 Simulating Continuous Distributions
    2.11 Further Proofs (Advanced)
    
  3. Expectation
    3.1 The Discrete Case
    3.2 The Absolutely Continuous Case
    3.3 Variance, Covariance and Correlation
    3.4 Generating Functions
       3.4.1 Characteristic Functions (Advanced)
    3.5 Conditional Expectation
       3.5.1 Discrete Case
       3.5.2 Absolutely Continuous Case
       3.5.3 Double Expectations
       3.5.4 Conditional Variance
    3.6 Inequalities
       3.6.1 Jensen's Inequality (Advanced)
    3.7 General Expectations (Advanced)
    3.8 Further Proofs (Advanced)
    
  4. Sampling Distributions and Limits
    4.1 Sampling Distributions
    4.2 Convergence in Probability
       4.2.1 The Weak Law of Large Numbers
    4.3 Convergence with Probability 1
       4.3.1 The Strong Law of Large Numbers
    4.4 Monte Carlo Approximations
    4.5 Convergence in Distributions
       4.5.1 The Central Limit Theorem
    4.6 Normal Distribution Theory
       4.6.1 The Chi-Square Distribution
       4.6.2 The t Distribution
       4.6.3 The F Distribution
    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
       5.4.1 Finite Population Sampling
       5.4.2 Random Sampling
       5.4.3 Histograms
       5.4.4 Survey Sampling
    5.5 Some Basic Inferences
       5.5.1 Descriptive Statistics
       5.5.2 Types of Inference
    
  6. Likelihood Inference
    6.1 The Likelihood Function
       6.1.1 Sufficient Statistics
    6.2 Maximum Likelihood Estimation
       6.2.1 The Multidimensional Case (Advanced)
    6.3 Inferences Based on the MLE
       6.3.1 Standard Errors and Bias
       6.3.2 Confidence Intervals
       6.3.3 Testing Hypotheses and P-values
       6.3.4 Sample Size Calculations: Length of Confidence Intervals
       6.3.5 Sample Size Calculations: Power
    6.4 Distribution-Free Models
       6.4.1 Method of Moments
       6.4.2 Bootstrapping
       6.4.3 The Sign Statistic and Inferences about Quatiles
    6.5 Large Sample Behavior of the MLE (Advanced)
    
  7. Bayesian inference
    7.1 The Prior and Posterior Distributions
    7.2 Inferences Based on the Posterior
       7.2.1 Estimation
       7.2.2 Credible Intervals
       7.2.3 Hypothesis Testing and Bayes Factors
    7.3 Bayesian Computations
       7.3.1 Asymptotic Normality of the Posterior
       7.3.2 Sampling from the Posterior
       7.3.3 Sampling from the Posterior Using Gibbs Sampling (Advanced)
    7.4 Choosing Priors
    7.5 Further Proofs (Advanced)
    
  8. Optimal Inferences
    8.1 Optimal Unbiased Estimation
       8.1.1 The Cramer-Rao Inequality (Advanced)
    8.2 Optimal Hypothesis Testing
       8.2.1 Likelihood Ratio Tests (Advanced)
    8.3 Optimal Bayesian Inferences
    8.4 Further Proofs (Advanced)
    
  9. Model Checking
    9.1 Checking the Sampling Model
       9.1.1 Residual Plots and Probability Plots
       9.1.2 The Chi-square Goodness of Fit Test
       9.1.3 Prediction and Cross-Validation
       9.1.4 What Do We Do When a Model Fails?
    9.2 Checking the Bayesian Model
    9.3 The Problem of Multiple Tests
    
  10. Relationships Among Variables
    10.1 Related Variables
       10.1.1 Cause-Effect Relationships
       10.1.2 Design for Experiments
    10.2 Categorical Response and Predictors
       10.2.1 Random Predictor
       10.2.2 Deterministic Predictor
       10.2.3 Bayesian Formulation
    10.3 Quantitative Response and Predictors
       10.3.1 The Method of Least Squares
       10.3.2 The Simple Linear Regression Model
       10.3.3 Bayesian Simple Linear Model (Advanced)
       10.3.4 The Multiple Linear Regression Model (Advanced)
    10.4 Quantitative Response and Categorical Predictors
       10.4.1 One Categorical Predictor (One-Way ANOVA)
       10.4.2 Repeated Measures (Paired Comparisons)
       10.4.3 Two Categorical Predictors (Two-Way ANOVA)
       10.4.4 Randomized Blocks
       10.4.5 One Categorical and Quantitative Predictor
    10.5 Categorical Response and Quantitative Predictors
    10.6 Further Proofs (Advanced)
    
  11. Advance Topic--Stochastic Processes
    11.1 Simple Random Walk
       11.1.1 The Distribution of the Fortune
       11.1.2 The Gambler's Ruin Problem
    11.2 Markov Chains
       11.2.1 Examples of Markov Chains
       11.2.2 Computing with Markov Chains
       11.2.3 Stationary Distributions
       11.2.4 Markov Chain Limit Theorem
    11.3 Markov Chain Monte Carlo
       11.3.1 The Metropolis-Hastings Algorithm
       11.3.2 The Gibbs Sampler
    11.4 Martingales
       11.4.1 Definition of a Martingale
       11.4.2 Expected Values
       11.4.3 Stopping Times
    11.5 Brownian Motion
       11.5.1 Faster and Faster Random Walks
       11.5.2 Brownian Motion as a Limit
       11.5.3 Diffusions and Stock Prices
    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
    A.6.1 Non-rectangular Regions
  B. Computations
  C. Common Distributions
  D. Tables
    D.1 Random Numbers
    D.2 Standard Normal Distributions
    D.3 Chi-square Distribution Probabilities
    D.4 Student Distribution Probabilities
    D.5 F Distribution Probabilities
    D.6 Binomial Distribution Probabilities
    
    Index