# Probability Foundations for Engineers

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## Book Description

Suitable for a first course in probability theory and designed specifically for industrial engineering and operations management students, **Probability Foundations for Engineers** covers theory in an accessible manner and includes numerous practical examples based on engineering applications. Essentially, everyone understands and deals with probability every day in their normal lives. Nevertheless, for some reason, when engineering students who have good math skills are presented with the mathematics of probability theory, there is a disconnect somewhere.

The book begins with a summary of set theory and then introduces probability and its axioms. The author has carefully avoided a theorem-proof type of presentation. He includes all of the theory but presents it in a conversational rather than formal manner, while relying on the assumption that undergraduate engineering students have a solid mastery of calculus. He explains mathematical theory by demonstrating how it is used with examples based on engineering applications. An important aspect of the text is the fact that examples are not presented in terms of "balls in urns". Many examples relate to gambling with coins, dice and cards but most are based on observable physical phenomena familiar to engineering students.

## Table of Contents

**Historical Perspectives**Formal Systems

Intuition

Exercises

**Introduction**

A Brief Review of Set Theory

A Brief Review of Set Theory

Definitions

Set Operations

Venn Diagrams

Dimensionality

Conclusion

Exercises

**Probability Basics**

Random Experiments, Outcomes, and Events

Probability

Probability Axioms

Conditional Probability

Independence

Exercises

**Random Variables and Distributions**

Random Variables

Distributions

Discrete Distribution Functions

Continuous Distribution Functions

Conditional Probability

Hazard Functions

Independent Random Variables

Exercises

**Joint, Marginal, and Conditional Distributions**

The Idea of Joint Random Variables

The Discrete Case

The Continuous Case

Independence

Bivariate and Multivariate Normal Distributions

Exercises

**Expectation**

Expectation and Functions of Random Variables

Expectation and Functions of Random Variables

Three Properties of Expectation

Expectation and Random Vectors

Conditional Expectation

General Functions of Random Variables

Expectation and Functions of Multiple Random Variables

Sums of Independent Random Variables

Exercises

**Construction of the Moment-Generating Function**

Moment-Generating Functions

Moment-Generating Functions

Convolutions

Joint Moment-Generating Functions

Conditional Moment-Generating Functions

Exercises

**Distribution-Free Approximations**

Approximations and Limiting Behavior

Approximations and Limiting Behavior

Normal and Poisson Approximations

Laws of Large Numbers and the Central Limit Theorem

Exercises

*Appendix: Cumulative Poisson Probabilities*

Index

Index

## Author(s)

### Biography

**Joel A. Nachlas** serves on the faculty of the Grado Department of Industrial and Systems Engineering at Virginia Tech. He has taught at Virginia Tech since 1974 and acts as the coordinator for the department’s graduate program in Operations Research. For the past twelve years, he has also taught Reliability Theory regularly at the Ecole Superiore d'Ingenieures de Nice-Sophia Antipolis.

Dr. Nachlas received his B. E. S. from the Johns Hopkins University in 1970, his M. S. in 1972 and his Ph. D. in 1976 both from the University of Pittsburgh. His research interests are in the applications of probability and statistics to problems in reliability and quality control. His work in microelectronics reliability has been performed in collaboration with and under the support of the IBM Corp., INTELSAT and the Bull Corp. He is the co-author of over fifty refereed articles, has served in numerous editorial and referee capacities and has lectured on reliability and maintenance topics throughout North America and Europe.

## Reviews

… responds to a need that I felt some years ago, which is to provide a basic and direct presentation of probability to engineers.— Enrico Zio, Politecnico di Milano, Dipartimento Energia, Milano Italy

I think this will make an excellent introductory book on probability for engineers and it will prepare the IE, CE and EE students for advanced courses that deal with random processes.— Edward A. Pohl, University of Arkansas, Fayetteville, USA

The theories are presented in a conversational rather than formal form as in most of the literature on probability. … introduces the reader in the field of randomness in a nice way. It gives a good starting point for more advanced studies. … creates a solid foundation to build up knowledge in more advanced statistical research. … The strength of the book is that it presents and translates the intuition concerning probability into mathematical structures using examples and explanations rather than traditional approach of theorem and proof. … perfect for undergraduate engineering students looking for a text book on probability.— Prof. Uday Kumar, Luleå University of Technology, SwedenOne of the distinctive feature (and one of its strength) of the book "probability foundations for engineers" is that it gives an in-depth and rigorous presentation of probability theory, while avoiding a classical mathematical – Theorems/Proofs- presentation. … As the author himself writes, he wants his book to be a supporting tool to go from intuition to mathematical rigor and this is certainly rewarding and fruitful from the pedagogical point of view. … The Approach of using everyday engineering intuition to introduce the basic notions of probabilities theory should make this book a valuable tool for engineering students who wants to learn the basic concepts and notions of probability theory and to be able to make use of these on engineering problems.— Christophe Bereguer, Grenoble Institute of Technology, France… this book takes a fresh approach to teaching undergraduate engineering students the fundamentals of probability. The book exploits students’ existing intuition regarding probabilistic concepts when presenting these concepts in a more rigorous manner. Students should be better able to retain the knowledge gained through reading this text because of the relevance of the examples and applications.—Lisa Maillart, University of Pittsburgh, Pennsylvania, USA