Proabability and Statistics: The Science of Uncert (Hardcover)

Michael J. Evans, Jeffrey S. Rosenthal

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商品描述

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

商品描述(中文翻譯)

與傳統的數學/統計入門教科書不同,《機率與統計:不確定性的科學》以將電腦融入課程和整合推論方法的現代風格為基礎。從一開始,本書就將模擬融入理論內容,並強調在整個過程中使用電腦計算。只需一年的微積分課程,數學和科學專業的學生就可以使用本書,體驗到應用和理論的融合,超越僅僅掌握技術性的要求。他們將獲得徹底的概率理論基礎,並進一步深入到統計推論理論及其應用。本書提出了一種整合的推論方法,包括頻率方法和貝葉斯方法。貝葉斯推論被發展為似然方法的邏輯擴展。一個單獨的章節專門介紹了模型檢查的重要主題,並應用於標準應用統計技術的背景中。全書中都提供了使用真實數據進行數據分析的示例。最後一章使用基本方法介紹了一些最重要的隨機過程模型。

目錄:
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