Proabability and Statistics: The Science of Uncert (Hardcover)

Michael J. Evans, Jeffrey S. Rosenthal



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.


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
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



1. 概率模型
1.1 概率:不確定性的度量
1.1.1 為什麼我們需要概率理論?
1.2 概率模型
1.3 概率模型的基本結果
1.4 有限空間上的均勻概率
1.4.1 組合原理
1.5 條件概率和獨立性
1.5.1 條件概率
1.5.2 事件的獨立性
1.6 P的連續性
1.7 進一步證明(高級)

2. 隨機變量和分佈
2.1 隨機變量
2.2 隨機變量的分佈
2.3 離散分佈
2.3.1 重要的離散分佈
2.4 連續分佈
2.4.1 重要的絕對連續分佈
2.5 累積分佈函數(cdf)
2.5.1 分佈函數的性質
2.5.2 離散分佈的cdf
2.5.3 絕對連續分佈的cdf
2.5.4 混合分佈
2.5.5 既非離散也非連續的分佈(高級)
2.6 一維隨機過程