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by Neil Weiss

Cover type: Print On DemandEdition: 06

Copyright: 2006

Publisher: Addison-Wesley Longman, Inc.

Published: 2006

International: No

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This text is intended primarily for a first course in mathematical probability for students in mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented students in the physical and social sciences. Prerequisite material consists of basic set theory and a firm foundation in elementary calculus, including infinite series, partial differentiation, and multiple integration. Some exposure to rudimentary linear algebra (e.g., matrices and determinants) is also desirable. This text includes pedagogical techniques not often found in books at this level, in order to make the learning process smooth, efficient, and enjoyable.

**Features**

- Illustrative examples. Definition and results (e.g., theorems and propositions) are followed by one or more examples that illustrate the concept or result in order to provide a concrete frame of reference.
- Abundant & varied exercises. The text contains an abundance of exercises, far more than most other probability books. The exercises provide an opportunity to vary the coverage and level.
- Applications. A diverse collection of applications appear throughout the text, some as examples or exercises, and others as entire sections. The last chapter of the text, Chapter 12, provides introductory materials--independent of one another--for three main follow-up courses: mathematical statistics, stochastic processes, and operations research.
- Biographies. Each chapter begins with a brief biography of a famous probabilist, mathematician, statistician, or scientist who has made substantial contributions to probability theory or its applications. These biographies help students obtain a perspective on how probability and its applications have developed.
- Motivation of key concepts. The importance of and rationale behind ideas such as the axioms of probability, conditional probability, independence, random variables, joint and conditional distributions, and expected value are made transparent. Formulas for probability mass functions and probability density functions are motivated instead of only stated. This helps students understand how such formulas arise naturally.

I. FUNDAMENTALS OF PROBABILITY.

1. Probability Basics.

Biography: Girolamo Cardano.

From Percentages to Probabilities.

Set Theory.

2. Mathematical Probability.

Biography: Andrei Kolmogorov.

Sample Space and Events.

Axioms of Probability.

Specifying Probabilities.

Basic Properties of Probability.

3. Combinatorial Probability.

Biography: James Bernoulli.

The Basic Counting Rule.

Permutations and Combinations.

Applications of Counting Rules to Probability.

4. Conditional Probability and Independence.

Biography: Thomas Bayes.

Conditional Probability.

The General Multiplication Rule.

Independent Events.

Bayes' Rule.

II. DISCRETE RANDOM VARIABLES.

5. Discrete Random Variables and Their Distributions.

Biography: Siméon-Dennis Poisson.

From Variables to Random Variables.

Probability Mass Functions.

Binomial Random Variables.

Hypergeometric Random Variables.

Poisson Random Variables.

Geometric Random Variables.

Other Important Discrete Random Variables.

Functions of a Discrete Random Variable.

6. Jointly Discrete Random Variables.

Biography: Blaise Pascal.

Joint and Marginal Probability Mass Functions: Bivariate Case.

Joint and Marginal Probability Mass Functions: Multivariate Case.

Conditional Probability Mass Functions.

Independent Random Variables.

Functions of Two or More Discrete Random Variables.

Sums of Discrete Random Variables.

7. Expected Value of Discrete Random Variables.

Biography: Christiaan Huygens.

From Averages to Expected Values.

Basic Properties of Expected Value.

Variance of Discrete Random Variables.

Variance, Covariance, and Correlation.

Conditional Expectation.

III. CONTINUOUS RANDOM VARIABLES.

8. Continuous Random Variables and Their Distributions.

Biography: Carl Friedrich Gauss.

Introducing Continuous Random Variables.

Cumulative Distribution Functions.

Probability Density Functions.

Uniform and Exponential Random Variables.

Normal Random Variables.

Other Important Continuous Random Variables.

Functions of a Continuous Random Variable.

9. Jointly Continuous Random Variables.

Biography: Pierre de Fermat.

Joint Cumulative Distribution Functions.

Introducing Joint Probability Density Functions.

Basic Properties of Joint Probability Density Functions.

Marginal and Conditional Probability Density Functions.

Independent Continuous Random Variables.

Functions of Two or More Continuous Random Variables.

Sums and Quotients of Continuous Random Variables.

Multidimensional Transformation Theorem.

10. Expected Value of Continuous Random Variables.

Biography: Pafnuty Chebyshev.

Expected Value of a Continuous Random Variable.

Basic Properties of Expected Value.

Variance, Covariance, and Correlation.

Conditional Expectation.

The Bivariate Normal Distribution.

IV. LIMIT THEOREMS AND ADVANCED TOPICS.

11. Generating Functions and Limit Theorems.

Biography: William Feller.

Moment Generating Functions.

Joint Moment Generating Functions.

Laws of Large Numbers.

The Central Limit Theorem.

12. Additional Topics.

Biography: Sir Ronald Fisher.

The Poisson Process.

Basic Queueing Theory.

The Multivariate Normal Distribution.

Sampling Distributions.

Appendices.

Index.

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Summary

This text is intended primarily for a first course in mathematical probability for students in mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented students in the physical and social sciences. Prerequisite material consists of basic set theory and a firm foundation in elementary calculus, including infinite series, partial differentiation, and multiple integration. Some exposure to rudimentary linear algebra (e.g., matrices and determinants) is also desirable. This text includes pedagogical techniques not often found in books at this level, in order to make the learning process smooth, efficient, and enjoyable.

**Features**

- Illustrative examples. Definition and results (e.g., theorems and propositions) are followed by one or more examples that illustrate the concept or result in order to provide a concrete frame of reference.
- Abundant & varied exercises. The text contains an abundance of exercises, far more than most other probability books. The exercises provide an opportunity to vary the coverage and level.
- Applications. A diverse collection of applications appear throughout the text, some as examples or exercises, and others as entire sections. The last chapter of the text, Chapter 12, provides introductory materials--independent of one another--for three main follow-up courses: mathematical statistics, stochastic processes, and operations research.
- Biographies. Each chapter begins with a brief biography of a famous probabilist, mathematician, statistician, or scientist who has made substantial contributions to probability theory or its applications. These biographies help students obtain a perspective on how probability and its applications have developed.
- Motivation of key concepts. The importance of and rationale behind ideas such as the axioms of probability, conditional probability, independence, random variables, joint and conditional distributions, and expected value are made transparent. Formulas for probability mass functions and probability density functions are motivated instead of only stated. This helps students understand how such formulas arise naturally.

Table of Contents

I. FUNDAMENTALS OF PROBABILITY.

1. Probability Basics.

Biography: Girolamo Cardano.

From Percentages to Probabilities.

Set Theory.

2. Mathematical Probability.

Biography: Andrei Kolmogorov.

Sample Space and Events.

Axioms of Probability.

Specifying Probabilities.

Basic Properties of Probability.

3. Combinatorial Probability.

Biography: James Bernoulli.

The Basic Counting Rule.

Permutations and Combinations.

Applications of Counting Rules to Probability.

4. Conditional Probability and Independence.

Biography: Thomas Bayes.

Conditional Probability.

The General Multiplication Rule.

Independent Events.

Bayes' Rule.

II. DISCRETE RANDOM VARIABLES.

5. Discrete Random Variables and Their Distributions.

Biography: Siméon-Dennis Poisson.

From Variables to Random Variables.

Probability Mass Functions.

Binomial Random Variables.

Hypergeometric Random Variables.

Poisson Random Variables.

Geometric Random Variables.

Other Important Discrete Random Variables.

Functions of a Discrete Random Variable.

6. Jointly Discrete Random Variables.

Biography: Blaise Pascal.

Joint and Marginal Probability Mass Functions: Bivariate Case.

Joint and Marginal Probability Mass Functions: Multivariate Case.

Conditional Probability Mass Functions.

Independent Random Variables.

Functions of Two or More Discrete Random Variables.

Sums of Discrete Random Variables.

7. Expected Value of Discrete Random Variables.

Biography: Christiaan Huygens.

From Averages to Expected Values.

Basic Properties of Expected Value.

Variance of Discrete Random Variables.

Variance, Covariance, and Correlation.

Conditional Expectation.

III. CONTINUOUS RANDOM VARIABLES.

8. Continuous Random Variables and Their Distributions.

Biography: Carl Friedrich Gauss.

Introducing Continuous Random Variables.

Cumulative Distribution Functions.

Probability Density Functions.

Uniform and Exponential Random Variables.

Normal Random Variables.

Other Important Continuous Random Variables.

Functions of a Continuous Random Variable.

9. Jointly Continuous Random Variables.

Biography: Pierre de Fermat.

Joint Cumulative Distribution Functions.

Introducing Joint Probability Density Functions.

Basic Properties of Joint Probability Density Functions.

Marginal and Conditional Probability Density Functions.

Independent Continuous Random Variables.

Functions of Two or More Continuous Random Variables.

Sums and Quotients of Continuous Random Variables.

Multidimensional Transformation Theorem.

10. Expected Value of Continuous Random Variables.

Biography: Pafnuty Chebyshev.

Expected Value of a Continuous Random Variable.

Basic Properties of Expected Value.

Variance, Covariance, and Correlation.

Conditional Expectation.

The Bivariate Normal Distribution.

IV. LIMIT THEOREMS AND ADVANCED TOPICS.

11. Generating Functions and Limit Theorems.

Biography: William Feller.

Moment Generating Functions.

Joint Moment Generating Functions.

Laws of Large Numbers.

The Central Limit Theorem.

12. Additional Topics.

Biography: Sir Ronald Fisher.

The Poisson Process.

Basic Queueing Theory.

The Multivariate Normal Distribution.

Sampling Distributions.

Appendices.

Index.

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 2006

International: No

Published: 2006

International: No

This text is intended primarily for a first course in mathematical probability for students in mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented students in the physical and social sciences. Prerequisite material consists of basic set theory and a firm foundation in elementary calculus, including infinite series, partial differentiation, and multiple integration. Some exposure to rudimentary linear algebra (e.g., matrices and determinants) is also desirable. This text includes pedagogical techniques not often found in books at this level, in order to make the learning process smooth, efficient, and enjoyable.

**Features**

- Illustrative examples. Definition and results (e.g., theorems and propositions) are followed by one or more examples that illustrate the concept or result in order to provide a concrete frame of reference.
- Abundant & varied exercises. The text contains an abundance of exercises, far more than most other probability books. The exercises provide an opportunity to vary the coverage and level.
- Applications. A diverse collection of applications appear throughout the text, some as examples or exercises, and others as entire sections. The last chapter of the text, Chapter 12, provides introductory materials--independent of one another--for three main follow-up courses: mathematical statistics, stochastic processes, and operations research.
- Biographies. Each chapter begins with a brief biography of a famous probabilist, mathematician, statistician, or scientist who has made substantial contributions to probability theory or its applications. These biographies help students obtain a perspective on how probability and its applications have developed.
- Motivation of key concepts. The importance of and rationale behind ideas such as the axioms of probability, conditional probability, independence, random variables, joint and conditional distributions, and expected value are made transparent. Formulas for probability mass functions and probability density functions are motivated instead of only stated. This helps students understand how such formulas arise naturally.

I. FUNDAMENTALS OF PROBABILITY.

1. Probability Basics.

Biography: Girolamo Cardano.

From Percentages to Probabilities.

Set Theory.

2. Mathematical Probability.

Biography: Andrei Kolmogorov.

Sample Space and Events.

Axioms of Probability.

Specifying Probabilities.

Basic Properties of Probability.

3. Combinatorial Probability.

Biography: James Bernoulli.

The Basic Counting Rule.

Permutations and Combinations.

Applications of Counting Rules to Probability.

4. Conditional Probability and Independence.

Biography: Thomas Bayes.

Conditional Probability.

The General Multiplication Rule.

Independent Events.

Bayes' Rule.

II. DISCRETE RANDOM VARIABLES.

5. Discrete Random Variables and Their Distributions.

Biography: Siméon-Dennis Poisson.

From Variables to Random Variables.

Probability Mass Functions.

Binomial Random Variables.

Hypergeometric Random Variables.

Poisson Random Variables.

Geometric Random Variables.

Other Important Discrete Random Variables.

Functions of a Discrete Random Variable.

6. Jointly Discrete Random Variables.

Biography: Blaise Pascal.

Joint and Marginal Probability Mass Functions: Bivariate Case.

Joint and Marginal Probability Mass Functions: Multivariate Case.

Conditional Probability Mass Functions.

Independent Random Variables.

Functions of Two or More Discrete Random Variables.

Sums of Discrete Random Variables.

7. Expected Value of Discrete Random Variables.

Biography: Christiaan Huygens.

From Averages to Expected Values.

Basic Properties of Expected Value.

Variance of Discrete Random Variables.

Variance, Covariance, and Correlation.

Conditional Expectation.

III. CONTINUOUS RANDOM VARIABLES.

8. Continuous Random Variables and Their Distributions.

Biography: Carl Friedrich Gauss.

Introducing Continuous Random Variables.

Cumulative Distribution Functions.

Probability Density Functions.

Uniform and Exponential Random Variables.

Normal Random Variables.

Other Important Continuous Random Variables.

Functions of a Continuous Random Variable.

9. Jointly Continuous Random Variables.

Biography: Pierre de Fermat.

Joint Cumulative Distribution Functions.

Introducing Joint Probability Density Functions.

Basic Properties of Joint Probability Density Functions.

Marginal and Conditional Probability Density Functions.

Independent Continuous Random Variables.

Functions of Two or More Continuous Random Variables.

Sums and Quotients of Continuous Random Variables.

Multidimensional Transformation Theorem.

10. Expected Value of Continuous Random Variables.

Biography: Pafnuty Chebyshev.

Expected Value of a Continuous Random Variable.

Basic Properties of Expected Value.

Variance, Covariance, and Correlation.

Conditional Expectation.

The Bivariate Normal Distribution.

IV. LIMIT THEOREMS AND ADVANCED TOPICS.

11. Generating Functions and Limit Theorems.

Biography: William Feller.

Moment Generating Functions.

Joint Moment Generating Functions.

Laws of Large Numbers.

The Central Limit Theorem.

12. Additional Topics.

Biography: Sir Ronald Fisher.

The Poisson Process.

Basic Queueing Theory.

The Multivariate Normal Distribution.

Sampling Distributions.

Appendices.

Index.