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# Probability and Statistics

## Second EditionMichael J. Evans; Jeffrey S. Rosenthal

©2011
E-book
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ISBN:9781429281270

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Unlike traditional introductory math/stat textbooks,

*Probability and Statistics: The Science of Uncertainty*brings a modern flavor to the course, incorporating the computer and offering an integrated approach to inference that includes the frequency approach and the Bayesian 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.The new edition includes a number of features designed to make the material more accessible and level-appropriate to the students taking this course today.

## E-book

Read online (or offline) with all the highlighting and notetaking tools you need to be successful in this course.

Learn More## Table of Contents

**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 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**