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Random Phenomena

Fundamentals of Probability and Statistics for Engineers

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## Book Description

Many of the problems that engineers face involve randomly varying phenomena of one sort or another. However, if characterized properly, even such randomness and the resulting uncertainty are subject to rigorous mathematical analysis.

Taking into account the uniquely multidisciplinary demands of 21st-century science and engineering, **Random Phenomena: Fundamentals of Probability and Statistics for Engineers** provides students with a working knowledge of how to solve engineering problems that involve randomly varying phenomena. Basing his approach on the principle of theoretical foundations before application, Dr. Ogunnaike presents a classroom-tested course of study that explains how to master and use probability and statistics appropriately to deal with uncertainty in standard problems and those that are new and unfamiliar.

Giving students the tools and confidence to formulate practical solutions to problems, this book offers many useful features, including:

As classic scientific boundaries continue to be restructured, the use of engineering is spilling over into more non-traditional areas, ranging from molecular biology to finance. This book emphasizes fundamentals and a "first principles" approach to deal with this evolution. It illustrates theory with practical examples and case studies, equipping readers to deal with a wide range of problems beyond those in the book.

**About the Author:**Professor Ogunnaike is Interim Dean of Engineering at the University of Delaware. He is the recipient of the 2008 American Automatic Control Council's Control Engineering Practice Award, the ISA's Donald P. Eckman Education Award, the Slocomb Excellence in Teaching Award, and was elected into the US National Academy of Engineering in 2012.

## Table of Contents

**Prelude**Approach Philosophy

Four Basic Principles

__I Foundations__

**Two Motivating Examples**

Yield Improvement in a Chemical Process

Quality Assurance in a Glass Sheet Manufacturing Process

Outline of a Systematic Approach

**Random Phenomena, Variability, and Uncertainty**

Two Extreme Idealizations of Natural Phenomena

Random Mass Phenomena

Introducing Probability

The Probabilistic Framework

__II Probability__

**Fundamentals of Probability Theory**

Building Blocks

Operations

Probability

Conditional Probability

Independence

**Random Variables and Distributions**

Distributions

Mathematical Expectation

Characterizing Distributions

Special Derived Probability Functions

**Multidimensional Random Variables**

Distributions of Several Random Variables

Distributional Characteristics of Jointly Distributed Random Variables

**Random Variable Transformations**

Single Variable Transformations

Bivariate Transformations

General Multivariate Transformations

**Application Case Studies I: Probability**

Mendel and Heredity

World War II Warship Tactical Response Under Attack

__III Distributions__

**Ideal Models of Discrete Random Variables**

The Discrete Uniform Random Variable

The Bernoulli Random Variable

The Hypergeometric Random Variable

The Binomial Random Variable

Extensions and Special Cases of the Binomial Random Variable

The Poisson Random Variable

**Ideal Models of Continuous Random Variables**

Gamma Family Random Variables

Gaussian Family Random Variables

Ratio Family Random Variables

**Information, Entropy, and Probability Models**

Uncertainty and Information

Entropy

Maximum Entropy Principles for Probability Modeling

Some Maximum Entropy Models

Maximum Entropy Models from General Expectations

**Application Case Studies II: In-Vitro Fertilization**

In-Vitro Fertilization and Multiple Births

Probability Modeling and Analysis

Binomial Model Validation

Problem Solution: Model-Based IVF Optimization and Analysis

Sensitivity Analysis

__IV Statistics__

**Introduction to Statistics**

From Probability to Statistics

Variable and Data Types

Graphical Methods of Descriptive Statistics

Numerical Descriptions

**Sampling**

The Distribution of Functions of Random Variables

Sampling Distribution of the Mean

Sampling Distribution of the Variance

**Estimation**

Criteria for Selecting Estimators

Point Estimation Methods

Precision of Point Estimates

Interval Estimates

Bayesian Estimation

**Hypothesis Testing**

Basic Concepts

Concerning Single Mean of a Normal Population

Concerning Two Normal Population Means

Determining β, Power, and Sample Size

Concerning Variances of Normal Populations

Concerning Proportions

Concerning Non-Gaussian Populations

Likelihood Ratio Tests

Discussion

**Regression Analysis**

Simple Linear Regression

"Intrinsically" Linear Regression

Multiple Linear Regression

Polynomial Regression

**Probability Model Validation**

Probability Plots

Chi-Squared Goodness-of-Fit Test

**Nonparametric Methods**

Single Population

Two Populations

Probability Model Validation

A Comprehensive Illustrative Example

**Design of Experiments**

Analysis of Variance

Single Factor Experiments

Two-Factor Experiments

General Multi-factor Experiments

2k Factorial Experiments and Design

Screening Designs: Fractional Factorial

Screening Designs: Plackett-Burman

1Response Surface Methodology

Introduction to Optimal Designs

**Application Case Studies III: Statistics**

Prussian Army Death-by-Horse Kicks

WW II Aerial Bombardment of London

US Population Dynamics: 1790–2000

Process Optimization

__V Applications__

**Reliability and Life Testing**

System Reliability

System Lifetime and Failure-Time Distributions

The Exponential Reliability Model

The Weibull Reliability Model

Life Testing

**Quality Assurance and Control**

Acceptance Sampling

Process and Quality Control

Chemical Process Control

Process and Parameter Design

**Introduction to Multivariate Analysis**

Multivariate Probability Models

Multivariate Data Analysis

Principal Components Analysis

**Appendix**

**Index**

## Reviews

The author does an excellent job presenting the material in an interesting way, making connections between theoretical and experimental statistics and between deterministic and probabilistic models. … a good book for engineers. … an excellent introductory mathematical statistics textbook for engineers. I like the fact that the theory is well developed throughout the chapters and that the transition between chapters is smooth. Compared with another introductory statistics resource for engineering (

Probability and Statistics for Engineering and the Sciencesby Jay Devore), I would choose this text … . I recommend this textbook with full confidence for engineering students who have the strong mathematical background, specifically differential and integral calculus. This book is distinguished from the crowded field by the well-explained theory of statistics and how it provides interesting applications. The big plus about this text is the variety and large amount of review questions, exercises, and application problems that the author provides, which in my opinion is crucial to the understanding of the theoretical concepts.

—Walid K. Sharabati,The American Statistician, August 2011This book offers many unique features in a crowded field of statistics books for engineers. ... The core concepts are written in an easy-to-understand format and students from various engineering disciplines can easily follow the theoretical concepts presented. The examples and application problems are selected from a wide range that spans catalysts in a chemical reactor,

in-vitrofertilization, molecular biology, reliability of parallel computer systems, population demographics, polymers, and finance. … I highly recommend this book for engineering students, professional engineers and applied statisticians dealing with systems involving random phenomena. Instructors who are looking for an alternative textbook should give a serious consideration to adopting this book.

—Ali Cinar,Technometrics, August 2011The theory is built up from real-life examples from which the first principles are deduced. This works well as it becomes immediately clear that these principles are relevant and useful in practical situations. … The book clearly explains both probabilistic and statistical concepts in minute detail and none of the essentials seem to be missing. All this is done without ever resorting to abstract mathematics, which is quite an achievement. … As an engineer involved in statistical data analysis, I would have loved to be taught from a book like this and I heartily recommend this book as a classroom textbook for both the clarity of the explanations and the amount of material covered. The book further accommodates such use with a large amount of review questions, exercises, application problems, and project assignments. However, the book is also suitable for self-study … . It will allow engineers who have to deal with statistics, but lack sufficient statistical background, to easily gain fundamental insights that are readily applicable in their working environment.

—Pieter Bastiaan Ober,Journal of Applied Statistics, 2011Introduces the theory by starting from well described engineering examples such that the resulting probability equations appear as the natural outcomes from engineering first principles and not as esoteric mathematics. Engineering significance is then reinforced with discussion of how the results apply to other problems… provide[s] an understanding of why and when statistical methods apply, and equally importantly, when pitfalls lurk. The continual relating of probability and statistics throughout the book is one of its strongest features. … Concepts are clearly explained. A good balance is struck between the providing critical theoretical underpinnings without overwhelming mathematical detail….

Examples from many engineering and science fields illustrate ideas and methods throughout the book, especially in the statistics material. …[presented] examples allow the reader to obtain a sense of the limitations of theory and methods and of the practical judgments required in applications to move to a problem resolution. A useful pedagogical feature is the repeated use of some data sets [on an accompanying CD], allowing students to see how new material provides new understanding.

Although aimed at the textbook market (several syllabus suggestions for 1 and 2-semester undergraduate and graduate courses are given in the Preface),Random Phenomenahas much to offer the industrial practitioner. As a chemical engineer who came to statistics out of industrial necessity and not from formal training or a career plan, I found new insights despite more than 20 years of practice, which includes providing internal statistics consulting and training

…all the fundamentals needed for further study in any of its topics are certainly provided. In summary,Random Phenomenais an excellent choice for anyone, educator or practitioner, wishing to impart or gain a fundamental understanding of probability and statistics from an engineering perspective.

—Dennis C. Williams, LyondellBasell Industries,The American Institute of Chemical Engineers Journal (AIChE Journal)