Introduction to Applied Probability

Introduction to Applied Probability

1st Edition - January 1, 1973

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  • Authors: Paul E. Pfeiffer, David A. Schum
  • eBook ISBN: 9781483277202

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Description

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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