# Introduction to Applied Probability

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Introduction to Applied Probability provides a basis for an intelligent application of probability ideas to a wide variety of phenomena for which it is suitable. It is intended as a tool for learning and seeks to point out and emphasize significant facts and interpretations which are frequently overlooked or confused by the beginner. The book covers more than enough material for a one semester course, enhancing the value of the book as a reference for the student. Notable features of the book are: the systematic handling of combinations of events (Section 3-5); extensive use of the mass concept as an aid to visualization; an unusually careful treatment of conditional probability, independence, and conditional independence (Section 6-4); the resulting clarification facilitates the formulation of many applied problems; the emphasis on events determined by random variables, which gives unity and clarity to many topics important for interpretation; and the utilization of the indicator function, both as a tool for dealing with events and as a notational device in the handling of random variables. Students of mathematics, engineering, biological and physical sciences will find the text highly useful.

## Table of Contents

Preface

Acknowledgments

Part I. Introduction

Chapter 1. An Approach to Probability

Introduction

1-1. Classical Probability

1-2. Toward a More General Theory

Chapter 2. Some Elementary Strategies of Counting

Introduction

2-1. Basic Principles

2-2. Arrangements

2-3. Binomial Coefficients

2-4. Verification of the Formulas for Arrangements

2-5. A Formal Representation of the Arrangement Problem

2-6. An Occupancy Problem Equivalent to the Arrangement Problem

2-7. Some Problems Utilizing Elementary Arrangements and Occupancy Situations as Component Operations

Problems

Part II. Basic Probability Model

Chapter 3. Sets and Events

Introduction

3-1. A Well-Defined Trial and Its Possible Outcomes

3-2. Events and the Occurrence of Events

3-3. Special Events and Compound Events

3-4. Classes of Events

3-5. Techniques for Handling Events

Problems

Chapter 4. A Probability System

Introduction

4-1. Requirements for a Formal Probability System

4-2. Basic Properties of a Probability System

4-3. Derived Properties of the Probability System

4-4. A Physical Analogy: Probability as Mass

4-5. Probability Mass Assignment on a Discrete Basic Space

4-6. On the Determination of Probabilities

4-7. Supplementary Examples

Problems

Chapter 5. Conditional Probability

Introduction

5-1. Conditioning and the Assignment of Probabilities

5-2. Some Properties of Conditional Probability

5-3. Supplementary Examples

5-4. Repeated Conditioning

5-5. Some Patterns of Inference

Problems

Chapter 6. Independence in Probability Theory

Introduction

6-1. The Defining Condition

6-2. Some Elementary Properties

6-3. Independent Classes of Events

6-4. Conditional Independence

6-5. Supplementary Examples

Problems

Chapter 7. Composite Trials and Sequences of Events

Introduction

7-1. Composite Trials

7-2. Repeated Trials

7-3. Bernoulli Trials

7-4. Sequences of Events

Problems

Part III. Random Variables

Chapter 8. Random Variables

Introduction

8-1. The Random Variable as a Function

8-2. Functions as Mappings

8-3. Events Determined by a Random Variable

8-4. The Indicator Function

8-5. Discrete Random Variables

8-6. Mappings and Inverse Images for Simple Random Variables

8-7. Mappings and Mass Transfer

8-8. Approximation by Simple Random Variables

Problems

Chapter 9. Distribution and Density Functions

Introduction

9-1. Some Introductory Examples

9-2. The Probability Distribution Function

9-3. Probability Mass and Density Functions

9-4. Additional Examples of Probability Mass Distributions

Problems

Chapter 10. Joint Probability Distributions

Introduction

10-1. Joint Mappings

10-2. Joint Distributions

10-3. Marginal Distributions

10-4. Properties of Joint Distribution Functions

10-5. Mass and Density Functions

10-6. Mixed Distributions

Problems

Chapter 11 Independence of Random Variables

Introduction

11-1. Definition and Examples

11-2. Independence and Probability Mass Distributions

11-3. A Simpler Condition for Independence

11-4. Independence Conditions for Distribution and Density Functions

Problems

Chapter 12. Functions of Random Variables

Introduction

12-1. Examples and Definition

12-2. Distribution and Mapping for a Function of a Single Random Variable

12-3. Functions of Two Random Variables

12-4. Independence of Functions of Random Variables

Problems

Part IV. Mathematical Expectation

Chapter 13. Mathematical Expectation and Mean Value

Introduction

13-1. The Concept

13-2. Fundamental Formulas

13-3. A Mechanical Interpretation

13-4. The Mean Value

13-5. Some General Properties

Problems

Chapter 14. Variance and Other Movements

Introduction

14-1. Definition and Interpretation of Variance

14-2. Some Properties of Variance

14-3. Variance for Some Common Distributions

14-4. Other Moments

14-5. Moment-Generating Function and Characteristic Function

14-6. Some Common Distributions

Problems

Chapter 15. Correlation and Covariance

Introduction

15-1. Joint Distributions for Centered and Standardized Random Variables

15-2. Characterization of the Joint Distributions

15-3. Covariance and the Correlation Coefficient

15-4. Linear Regression

15-5. Additional Interpretations of p

15-6. Linear Transformations of Uncorrelated Random Variables

Problems

Chapter 16. Conditional Expectation

Introduction

16-1. Averaging Over a Conditioning Event

16-2. A Conditioning Event Determined by a Second Random Variable

16-3. Averaging Over a Partition of an Event

16-4. Conditioning by a Discrete Random Variable

16-5. Conditioning by a Continuous Random Variable

16-6. Some Properties of Conditional Expectation

16-7. Regression Theory

16-8. Estimating a Probability

Problems

Part V. Sequences Of Random Variables

Chapter 17. Sequences of Random Variables

Introduction

17-1. Composite Trials

17-2. The Multinomial Distribution

17-3. The Law of Large Numbers

17-4. The Strong Law of Large Numbers

17-5. The Central Limit Theorem

17-6. Applications to Statistics

Problems

Chapter 18. Constant Markov Chains

Introduction

18-1. Definitions and an Introductory Example

18-2. Some Examples of Markov Chains

18-3. Transition Diagrams and Accessibility of States

18-4. Recurrence and Periodicity

18-5. Some Results for Irreducible Chains

Problems

Appendix A. Numerical Tables

A-1. Factorials and Their Logarithms

A-2. The Exponential Function

A-3. Binomial Coefficients

A-4. The Summed Binomial Distribution

A-5. Standardized Normal Distribution Function

Appendix B. Some Mathematical Aids

B-1. Binary Representation of Numbers

B-2. Geometric Series

B-3. Extended Binomial Coefficient

B-4. Gamma Function

B-5. Beta Function

B-6. Matrices

Selected References

Selected Answers and Hints

Index of Symbols and Abbreviations

Index

## Product details

- No. of pages: 420
- Language: English
- Copyright: © Academic Press 1973
- Published: January 1, 1973
- Imprint: Academic Press
- eBook ISBN: 9781483277202

## About the Authors

### Paul E. Pfeiffer

### David A. Schum

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