# An Introduction to Probability and Statistical Inference

## 2nd Edition

**Authors:**George Roussas

**eBook ISBN:**9780128004371

**Hardcover ISBN:**9780128001141

**Imprint:**Academic Press

**Published Date:**25th September 2014

**Page Count:**624

## Description

*An Introduction to Probability and Statistical Inference, Second Edition,* guides you through probability models and statistical methods and helps you to think critically about various concepts. Written by award-winning author George Roussas, this book introduces readers with no prior knowledge in probability or statistics to a thinking process to help them obtain the best solution to a posed question or situation. It provides a plethora of examples for each topic discussed, giving the reader more experience in applying statistical methods to different situations.

This text contains an enhanced number of exercises and graphical illustrations where appropriate to motivate the reader and demonstrate the applicability of probability and statistical inference in a great variety of human activities. Reorganized material is included in the statistical portion of the book to ensure continuity and enhance understanding. Each section includes relevant proofs where appropriate, followed by exercises with useful clues to their solutions. Furthermore, there are brief answers to even-numbered exercises at the back of the book and detailed solutions to all exercises are available to instructors in an Answers Manual.

This text will appeal to advanced undergraduate and graduate students, as well as researchers and practitioners in engineering, business, social sciences or agriculture.

## Key Features

- Content, examples, an enhanced number of exercises, and graphical illustrations where appropriate to motivate the reader and demonstrate the applicability of probability and statistical inference in a great variety of human activities
- Reorganized material in the statistical portion of the book to ensure continuity and enhance understanding
- A relatively rigorous, yet accessible and always within the prescribed prerequisites, mathematical discussion of probability theory and statistical inference important to students in a broad variety of disciplines
- Relevant proofs where appropriate in each section, followed by exercises with useful clues to their solutions
- Brief answers to even-numbered exercises at the back of the book and detailed solutions to all exercises available to instructors in an Answers Manual

## Readership

Advanced undergraduateand graduate students, researchers and practitioners in engineering, business, social sciences or agriculture

## Table of Contents

- Dedication
- Preface
- Overview
- Chapter Descriptions
- Features
- Brief Preface of the Revised Version
- Acknowledgments and Credits

- Chapter 1: Some motivating examples and some fundamental concepts
- 1.1 Some Motivating Examples
- 1.2 Some Fundamental Concepts
- 1.3 Random Variables

- Chapter 2: The concept of probability and basic results
- 2.1 Definition of Probability and Some Basic Results
- 2.2 Distribution of a Random Variable
- 2.3 Conditional Probability and Related Results
- 2.4 Independent Events and Related Results
- 2.5 Basic Concepts and Results in Counting

- Chapter 3: Numerical characteristics of a random variable, some special random variables
- 3.1 Expectation, Variance, and Moment Generating Function of a Random Variable
- 3.2 Some Probability Inequalities
- 3.3 Some Special Random Variables
- 3.4 Median and Mode of a Random Variable

- Chapter 4: Joint and conditional p.d.f.’s, conditional expectation and variance, moment generating function, covariance, and correlation coefficient
- 4.1 Joint D.F. and Joint p.d.f. of Two Random Variables
- 4.2 Marginal and Conditional p.d.f.'s, Conditional Expectation and Variance
- 4.3 Expectation of a Function of Two r.v.'s, Joint and Marginal m.g.f.'s, Covariance, and Correlation Coefficient
- 4.4 Some Generalizations to
*k*Random Variables - 4.5 The Multinomial, the Bivariate Normal, and the Multivariate Normal Distributions

- Chapter 5: Independence of random variables and some applications
- 5.1 Independence of Random Variables and Criteria of Independence
- 5.2 The Reproductive Property of Certain Distributions

- Chapter 6: Transformation of random variables
- 6.1 Transforming a Single Random Variable
- 6.2 Transforming Two or More Random Variables
- 6.3 Linear Transformations
- 6.4 The Probability Integral Transform
- 6.5 Order Statistics

- Chapter 7: Some modes of convergence of random variables, applications
- 7.1 Convergence in Distribution or in Probability and Their Relationship
- 7.2 Some Applications of Convergence in Distribution: WLLN and CLT
- 7.3 Further Limit Theorems

- Chapter 8: An overview of statistical inference
- 8.1 The Basics of Point Estimation
- 8.2 The Basics of Interval Estimation
- 8.3 The Basics of Testing Hypotheses
- 8.4 The Basics of Regression Analysis
- 8.5 The Basics of Analysis of Variance
- 8.6 The Basics of Nonparametric Inference

- Chapter 9: Point estimation
- 9.1 Maximum Likelihood Estimation: Motivation and Examples
- 9.2 Some Properties of MLE's
- 9.3 Uniformly Minimum Variance Unbiased Estimates
- 9.4 Decision-Theoretic Approach to Estimation
- 9.5 Other Methods of Estimation

- Chapter 10: Confidence intervals and confidence regions
- 10.1 Confidence Intervals
- 10.2 Confidence Intervals in The Presence of Nuisance Parameters
- 10.3 A Confidence Region for (
*μ*,*σ*2) in the*N*(*μ*,*σ*2) Distribution - 10.4 Confidence Intervals with Approximate Confidence Coefficient

- Chapter 11: Testing hypotheses
- 11.1 General Concepts, Formulation of Some Testing Hypotheses
- 11.2 Neyman-Pearson Fundamental Lemma, Exponential Type Families, UMP Tests for Some Composite Hypotheses
- 11.3 Some Applications of Theorems 2
- 11.4 Likelihood Ratio Tests

- Chapter 12: More about testing hypotheses
- 12.1 Likelihood Ratio Tests in the Multinomial Case and Contingency Tables
- 12.2 A Goodness-of-Fit Test
- 12.3 Decision-Theoretic Approach to Testing Hypotheses
- 12.4 Relationship between Testing Hypotheses and Confidence Regions

- Chapter 13: A simple linear regression model
- 13.1 Setting up The Model—The Principle of Least Squares
- 13.2 The Least Squares Estimates of
*β*1 and*β*2 and Some of Their Properties - 13.3 Normally Distributed Errors: MLE's of
*β*1,*β*2, and*σ*2, Some Distributional Results - 13.4 Confidence Intervals and Hypotheses Testing Problems
- 13.5 Some Prediction Problems
- 13.6 Proof of Theorem 5
- 13.7 Concluding Remarks

- Chapter 14: Two models of analysis of variance
- 14.1 One-Way Layout with the Same Number of Observations Per Cell

- Chapter 15: Some topics in nonparametric inference
- 15.1 Some Confidence Intervals with Given Approximate Confidence Coefficient
- 15.2 Confidence Intervals for Quantiles of a Distribution Function
- 15.3 The Two-Sample Sign Test
- 15.4 The Rank Sum and the Wilcoxon–Mann–Whitney Two-Sample Tests
- 15.5 Nonparametric Curve Estimation

- Tables
- Some notation and abbreviations
- Answers to even-numbered exercises
- Chapter 1
- Chapter 2
- Chapter 3
- Chapter 4
- Chapter 5
- Chapter 6
- Chapter 7
- Chapter 9
- Chapter 10
- Chapter 11
- Chapter 12
- Chapter 13
- Chapter 14
- Chapter 15

- Index
- Inside Back Matter

## Details

- No. of pages:
- 624

- Language:
- English

- Copyright:
- © Academic Press 2015

- Published:
- 25th September 2014

- Imprint:
- Academic Press

- eBook ISBN:
- 9780128004371

- Hardcover ISBN:
- 9780128001141

## About the Author

### George Roussas

George G. Roussas earned a B.S. in Mathematics with honors from the University of Athens, Greece, and a Ph.D. in Statistics from the University of California, Berkeley. As of July 2014, he is a Distinguished Professor Emeritus of Statistics at the University of California, Davis. Roussas is the author of five books, the author or co-author of five special volumes, and the author or co-author of dozens of research articles published in leading journals and special volumes. He is a Fellow of the following professional societies: The American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), The Royal Statistical Society (RSS), the American Association for the Advancement of Science (AAAS), and an Elected Member of the International Statistical Institute (ISI); also, he is a Corresponding Member of the Academy of Athens. Roussas was an associate editor of four journals since their inception, and is now a member of the Editorial Board of the journal Statistical Inference for Stochastic Processes. Throughout his career, Roussas served as Dean, Vice President for Academic Affairs, and Chancellor at two universities; also, he served as an Associate Dean at UC-Davis, helping to transform that institution's statistical unit into one of national and international renown. Roussas has been honored with a Festschrift, and he has given featured interviews for the Statistical Science and the Statistical Periscope. He has contributed an obituary to the IMS Bulletin for Professor-Academician David Blackwell of UC-Berkeley, and has been the coordinating editor of an extensive article of contributions for Professor Blackwell, which was published in the Notices of the American Mathematical Society and the Celebratio Mathematica.

### Affiliations and Expertise

University of California, Davis, USA