Written by two leading statisticians, this applied introduction to the mathematics of probability and statistics emphasizes the existence of variation in almost every process, and how the study of probability and statistics helps us understand this variation. Designed for students with a background in calculus, this book continues to reinforce basic mathematical concepts with numerous real-world examples and applications to illustrate the relevance of key concepts.

• The included CD-ROM contains all of the data sets in a variety of formats for use with most statistical software packages. This disc also includes several applications of Minitab^® and Maple^™.
• Historical vignettes at the end of each chapter outline the origin of the greatest accomplishments in the field of statistics, adding enrichment to the course.

 Content updates

• The first five chapters have been reorganized to cover a standard probability course with more real examples and exercises. These chapters are important for students wishing to pass the first actuarial exam, and cover the necessary material needed for students taking this course at the junior level.
• Chapters 6 and 7 on estimation and tests of statistical hypotheses tie together confidence intervals and tests, including one-sided ones. There are separate chapters on nonparametric methods, Bayesian methods, and Quality Improvement.
• Chapters 4 and 5 include a strong discussion on conditional distributions and functions of random variables, including Jacobians of transformations and the moment-generating technique. Approximations of distributions like the binomial and the Poisson with the normal can be found using the central limit theorem.
• Chapter 8 (Nonparametric Methods) includes most of the standards tests such as those by Wilcoxon and also the use of order statistics in some distribution-free inferences.
• Chapter 9 (Bayesian Methods) explains the use of the "Dutch book" to prove certain probability theorems.
• Chapter 11 (Quality Improvement) stresses how important W. Edwards Deming's ideas are in understanding variation and how they apply to everyday life.

TABLE OF CONTENTS: 

Preface

Prologue

1. Probability

1.1 Basic Concepts

1.2 Properties of Probability

1.3 Methods of Enumeration

1.4 Conditional Probability

1.5 Independent Events

1.6 Bayes's Theorem

2. Discrete Distributions

2.1 Random Variables of the Discrete Type

2.2 Mathematical Expectation

2.3 The Mean, Variance, and Standard Deviation

2.4 Bernoulli Trials and the Binomial Distribution

2.5 The Moment-Generating Function

2.6 The Poisson Distribution

3. Continuous Distributions

3.1 Continuous-Type Data

3.2 Exploratory Data Analysis

3.3 Random Variables of the Continuous Type

3.4 The Uniform and Exponential Distributions

3.5 The Gamma and Chi-Square Distributions

3.6 The Normal Distribution

3.7 Additional Models

4. Bivariate Distributions

4.1 Distributions of Two Random Variables

4.2 The Correlation Coefficient

4.3 Conditional Distributions

4.4 The Bivariate Normal Distribution

5. Distributions of Functions of Random Variables

5.1 Functions of One Random Variable

5.2 Transformations of Two Random Variables

5.3 Several Independent Random Variables

5.4 The Moment-Generating Function Technique

5.5 Random Functions Associated with Normal Distributions

5.6 The Central Limit Theorem

5.7 Approximations for Discrete Distributions

6. Estimation

6.1 Point Estimation

6.2 Confidence Intervals for Means

6.3 Confidence Intervals for Difference of Two Means

6.4 Confidence Intervals for Variances

6.5 Confidence Intervals for Proportions

6.6 Sample Size.

6.7 A Simple Regression Problem

6.8 More Regression

7. Tests of Statistical Hypotheses

7.1 Tests about Proportions

7.2 Tests about One Mean

7.3 Tests of the Equality of Two Means

7.4 Tests for Variances

7.5 One-Factor Analysis of Variance

7.6 Two-Factor Analysis of Variance

7.7 Tests Concerning Regression and Correlation

8. Nonparametric Methods

8.1 Chi-Square Goodness of Fit Tests

8.2 Contingency Tables

8.3 Order Statistics

8.4 Distribution-Free Confidence Intervals for Percentiles

8.5 The Wilcoxon Tests

8.6 Run Test and Test for Randomness

8.7 Kolmogorov-Smirnov Goodness of Fit Test

8.8 Resampling Methods

9. Bayesian Methods

9.1 Subjective Probability

9.2 Bayesian Estimation

9.3 More Bayesian Concepts

10. Some Theory

10.1 Sufficient Statistics

10.2 Power of a Statistical Test

10.3 Best Critical Regions

10.4 Likelihood Ratio Tests

10.5 Chebyshev's Inequality and Convergence in Probability

10.6 Limiting Moment-Generating Functions

10.7 Asymptotic Distributions of Maximum Likelihood Estimators

11. Quality Improvement Through Statistical Methods

11.1 Time Sequences

11.2 Statistical Quality Control

11.3 General Factorial and 2^k Factorial Designs

11.4 Understanding Variation

A. Review of Selected Mathematical Techniques

      A.1 Algebra of Sets

      A.2 Mathematical Tools for the Hypergeometric Distribution

      A.3 Limits

      A.4 Infinite Series

      A.5 Integration

      A.6 Multivariate Calculus

B. References

C. Tables

D. Answers to Odd-Numbered Exercises