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Mathematics

Probability Demystified by Allan G. Bluman (McGraw-Hill Professional) Don't Roll The Dice Learning Probability!

Now anyone who ever flipped a coin, played cards, or placed a bet can grasp the principles that govern probability -- without formal training, unlimited time, or an Einstein IQ. In Probability Demystified, experienced math instructor Allan Bluman provides an illuminating and entertaining way to master chance, odds, and predictability.

With Probability Demystified, you learn the subject one simple step at a time -- at your own speed. This unique, self-teaching guide offers problems at the end of each chapter and section to pinpoint weaknesses, and a 60-question "final exam" to reinforce the entire book.

If you have basic math and a little algebra, you can master probability. This fast and entertaining self-teaching course makes it fun and easy to:

• Learn probability theory with examples using familiar items such as coins, cards, and dice

• Discover how actuaries predict the likelihood of events

• Play games of chance with deeper understanding

• Construct a solid foundation for statistics

• Be better prepared for standardized tests

• Take a "final exam" and grade it yourself!

Simple enough for beginners and students but always challenging, Probability Demystified is the most interesting self-teaching tool or brush-up you can find. 

Probability for Electrical and Computer Engineers by Charles W. Therrien, Murali Tummala (CRC Press) Written specifically for electrical and computer engineers, this book provides an introduction to probability and random variables. It includes methods of probability that deal with computing the likelihood of uncertain events for scientists and engineers who predict the outcome of experiments, extrapolate results from a small case to a larger one, and design systems that will perform optimally when the exact characteristics of the inputs are unknown. Electrical and computer engineers seeking solutions to practical problems will find it a valuable resource in the design of communication systems, control systems, military or medical sensing or monitoring systems, and computer networks.

The authors believe that the topics in this book should be taught at the earliest possible level. In most universities, this would represent the junior or senior undergraduate level. Although it is possible for students to study this material at an even earlier (e.g., sophomore) level, in this case students may have less appreciation of the applications and the instructor may want to focus more on the theory and only Chapters 1 through 6.

Ideally we feel that a course based on this book should directly precede a more advanced course on communications or random processes for students studying those areas, as it does at the Naval Postgraduate School. While several engineering books devoted to the more advanced aspects of random processes and systems also include background material on probability, we feel that a book that focuses mainly on probability for engineers with engineering applications has been long overdue.

While they have written this book for students of electrical and computer engineering, we do not mean to exclude students studying other branches of engineering or physical sciences. Indeed, the theory is most relevant to these other disciplines, and the applications and examples, although drawn from our own discipline, are ubiquitous in other areas.

After an introductory chapter, the text begins in Chapter 2 with the algebra of events and probability. Considerable emphasis is given to representation of the sample space for various types of problems. We then move on to random variables, discrete and continuous, and transformations of random variables in Chapter 3. Expectation is considered to be sufficiently important that a separate chapter (4) is devoted to the topic. This chapter also provides a treatment of moments and generating functions. Chapter 5 deals with two and more random variables and includes a short (more advanced) introduction to random vectors for instructors with students having sufficient background in linear algebra who may want to cover this topic. Chapter 6 groups together topics of convergence, limit theorems, and parameter estimation. Some of these topics are typically taught at a more advanced level, but we feel that introducing them at this more basic level and relating them to earlier topics in the text helps students appreciate the need for a strong theoretical basis to support practical engineering applications.

Chapter 7 provides a brief introduction to random processes and linear systems while

Chapter 8 deals with discrete and continuous Markov processes and an introduction to queueing theory. Either or both of these chapters may be skipped for a basic level course in probability; however we have found that both are useful even for students who later plan to take a more advanced course in random processes.

Applications and examples are distributed throughout the text. Moreover, the authors have tried to introduce the applications at the earliest opportunity, as soon as the supporting probabilistic topics have been covered.

The overview of the probabilistic model in this chapter is meant to depict the general framework of ideas that are relevant for this area of study and to put some of these concepts in perspective. The set of examples is to convince you that this topic, which is basically an area of mathematics, is extremely important for engineers.

In the chapters that follow is presented the study of probability in a context that continues to show its importance for engineering applications. All of the applications cited above and others are discussed explicitly in later chapters. Our goal has been to bring in the applications as early as possible even if this requires considerable simplification to a real world problem. We have also tried to keep the discussion light (although not totally lacking in rigor) and to occasionally inject some light humor.

Next begins with a discussion of the probability model, events, and probability measure. This is followed in Chapter 3 by a discussion of random variables. Averages, known as statistical expectation, form an important part of the theory and are discussed in a fairly short Chapter 4. Chapter 5 then deals with multiple random variables, expectation, and random vectors. We have left the topic of theorems, bounds, and estimation to fairly late in the book, in Chapter 6. While these topics are important, we want to present the more application-oriented material first. Chapter 7 then provides an introduction to random processes and the final chapter (Chapter 8) continues to discuss the special classes of random processes that pertain to the topic of queueing.

Topics in mathematics are not always easy especially when old paradigms are broken and new concepts need to be developed. Perseverance leads to success however, and success leads to enjoyment. We hope that you will come to enjoy this topic as well as we have enjoyed teaching and writing about it.

 

An Introduction to Probability and Inductive Logic by Ian Hacking (Cambridge University Press) (PAPERBACK) This is an introductory textbook on probability and induction written by one of the world's fore­most philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students ( not only those majoring in philosophy) and assumes no formal train­ing in elementary symbolic logic. It offers a comprehensive course covering all basic definitions of of induction and probability, and it considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction.

The key features of the book are: a lively and vigorous prose style, lucid and systematic organization and presentation of the ideas; many practical applications; a rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science; numerous brief historical accounts of how fundamental ideas of probability and induction developed; a full bibliography of further reading

Although designed primarily for courses in philosophy, the book could certainly be read and enjoyed by those in the social sciences (particularly psychology, economics, political science and sociology) or medical sciences (such as epidemiology) seeking a reader-friendly account of the basic ideas of probability and induction.

"While written as an introductory text, it is full of philosophical wisdom. Moreover, this is wis­dom that most students of philosophy need but find very hard to acquire. Hacking explains all the basic ideas of probability theory, the philosophical puzzles they raise, the standard lines of response, their strengths and weaknesses. He writes with the authority of someone who has helped form the debates and understands everything properly, but at the same time he gives a fair hearing to all positions worth taking seriously.

"At some point in the career of most philosophy students, graduates and undergraduates alike, they read stuff, which uses probabilistic ideas and turn to their teachers for guidance. I can imagine that the teachers' automatic response for some decades to come will be to send these students to Hacking."            - David Papineau, King's College, London

"Hacking's textbook is likely to become the standard for inductive logic courses. He writes sim­ply, in a lively style, without oversimplification... . Lively and original examples drawn from everyday life create the appropriate context to prepare students to think critically about the barrage of statistical arguments that confront us on a daily basis... . Hacking's textbook sheds much needed light on the mystique of statistical reasoning."

- Katherine van Uum, Grinnell College, Iowa

Inductive logic is unlike deductive or symbolic logic. In deductive reasoning, when you have true premises and a valid argument, the conclusion must be true too. Valid deductive arguments do not take risks.

Inductive logic takes risks. You can have true premises, a good argument, but a false conclusion. Inductive logic uses probability to analyse that kind of risky argument.

Good News

Inductive reasoning is a guide in life. People make risky decisions all the time. It plays a much larger part in everyday affairs than deductive reasoning.

Bad News

People are very bad when reasoning about risks. We make a lot of mistakes when we use probabilities.

This book starts with a list of seven Odd Questions. They look pretty simple. But most people get some of the answers wrong. The last group of nine-year­olds I tested did better than a group of professors. Try the Odd Questions. Each one is discussed later in the book.

Practical Aims

This book can help you understand, use, and act on probabilities, risks, and statistics. We live our lives taking chances, acting when we don't know enough. Every day we experience a lot of uncertainties. This book is about the kinds of actions you can take when you are uncertain what to do. It is about the inferences you can draw when your evidence leaves you unsure what is true.

We Are Drowning in Probabilities and Statistics

Nowadays you can't escape hearing about probabilities, statistics, and risk. Everything-jobs, sex, war, health, sport, grades, the environment, politics, as­tronomy, genetics-is wrapped up in probabilities.

This is new. If your grandparents lived in North America they seldom came

Lux beauty soap" (a famous line on a weekly radio show). Now we get polls, surveys, and digests of opinion all the time. No public decision can be made without statistical analysis, risk analysis, environmental impact reports.

It is pretty hard to understand what all the numbers mean. This book aims at helping you understand them. How to use them. How they are abused. When inductive reasoning is fallacious or uses sloppy rhetoric. How people get fooled by numbers. How numbers are often used to conceal ignorance. How not to be conned.

There is a famous problem in philosophy called the problem of induction. That comes at the end of the book.

There are ethical questions about risk. Some philosophers say we should always act so as to maximize the common good. Others say that duty and right and wrong come before cost-benefit thinking. These questions arise in Chapter 9.

There are even some probability arguments for, and against, religious belief. One comes up in Chapter 10.

There are philosophical arguments about probability itself. Right now there are big disagreements over the basic ideas of inductive inference. Different schools of thought approach practical issues in different ways. Most beginning statistics courses pretend that there is no disagreement. This is a philosophy book, so it puts the competing ideas up front. It tries to be fair to all parties. Calculation

To get a grip on chances, risks, or probabilities, you need numbers. But even if you hate calculating you can use this book. Don't be put off by the formulas. This book is about ideas that we represent by numbers. A philosophy book is concerned with the ideas behind calculations. It is not concerned with computing precise solutions to complicated problems.

You do not need a pocket calculator for most of the exercises, because the numbers usually "cancel" for an easy solution. Students who learn not to use calculators solve most of the problems more quickly than students who use them. Gambling

Many simple examples of probability mention games of chance. You may not like this. People have different attitudes toward gambling for money. Some think it is fun. Some are addicted to it. Some find it boring. Many people think it is immoral. Governments all over the world love legalized forms of gambling such as lotteries, because they are an easy way to produce extra revenue. Gamblers, as a group, always lose, and lose a lot. This book is not an advertisement for gambling. Quite the opposite! Aside from friendly occasions-a bet on the ball game, or a late night poker party among friends-gambling is a waste of time, money, and human dignity.

Nevertheless, in our risky lives we "gamble" all the time. We make decisions

because we do not know enough. Models based on games help us to understand these decisions and inferences. They can clarify the ways in which we think about chance.

That is why we so often turn to dice and other randomizers used in betting. They crop up in the Odd Questions. Yet they soon lead to practical issues like testimony in court (Odd Question 5) and medical diagnosis (Odd Question 6).

Try your luck at these questions, without any calculating. Each question will be discussed in the text. Do not be surprised if you make mistakes!

Foundations of Modern Probability by Olav Kallenberg (Probability and Its Applications: Springer) is unique for its broad and yet comprehensive coverage of modern probability theory, ranging from first principles and standard textbook material to more advanced topics. In spite of the economical exposition, careful proofs are provided for all main results. After a detailed discussion of classical limit theorems, martingales, Markov chains, random walks, and stationary processes, the author moves on to a modern treatment of Brownian motion, L vy processes, weak convergence, It calculus, Feller processes, and SDEs. The more advanced parts include material on local time, excursions, and additive functionals, diffusion processes, PDEs and potential theory, predictable processes, and general semimartingales. Though primarily intended as a general reference for researchers and graduate students in probability theory and related areas of analysis, the book is also suitable as a text for graduate and seminar courses on all levels, from elementary to advanced. Numerous easy to more challenging exercises are provided, especially for the early chapters.

Foundations of Modern Probability is generally only useful at a graduate level. It is not written for the people who have not had senior level advanced probability courses. It concision and abstractness makes it a useful reference. Its proofs are elegant but lack much detail, while statements are generally abstract and too concise to not need explanation. Still there are few books about the foundations of probability that tries to cover the essentials in such a comprehensive manner as to make it if not an ideal textbook then an ideal reference work.

Author’s Preface: Some thirty years ago it was still possible, as Loeve so ably demonstrated, to write a single book in probability theory containing practically everything worth knowing in the subject. The subsequent development has been explosive, and today a corresponding comprehensive coverage would require a whole library. Researchers and graduate students alike seem compelled to a rather extreme degree of specialization. As a result, the subject is threatened by disintegration into dozens or hundreds of subfields.

At the same time the interaction between the areas is livelier than ever, and there is a steadily growing core of key results and techniques that every probabilist needs to know, if only to read the literature in his or her own field. Thus, it seems essential that we all have at least a general overview of the whole area, and we should do what we can to keep the subject together. The present volume is an earnest attempt in that direction.

My original aim was to write a book about "everything." Various space and time constraints forced me to accept more modest and realistic goals for the project. Thus, "foundations" had to be understood in the narrower sense of the early 1970s, and there was no room for some of the more recent developments. I especially regret the omission of topics such as large deviations, Gibbs and Palm measures, interacting particle systems, stochastic differential geometry, Malliavin calculus, SPDEs, measure‑valued diffusions, and branching and superprocesses. Clearly plenty of fundamental and intriguing material remains for a possible second volume.

Even with my more limited, revised ambitions, I had to be extremely selective in the choice of material. More importantly, it was necessary to look for the most economical approach to every result I did decide to include. In the latter respect, I was surprised to see how much could actually be done to simplify and streamline proofs, often handed down through generations of textbook writers. My general preference has been for results conveying some new idea or relationship, whereas many propositions of a more technical nature have been omitted. In the same vein, I have avoided technical or computational proofs that give little insight into the proven results. This conforms with my conviction that the logical structure is what matters most in mathematics, even when applications is the ultimate goal.

Though the book is primarily intended as a general reference, it should also be useful for graduate and seminar courses on different levels, ranging from elementary to advanced. Thus, a first‑year graduate course in measure theoretic probability could be based on the first ten or so chapters, while the rest of the book will readily provide material for more advanced courses on various topics. Though the treatment is formally self‑contained, as far as measure theory and probability are concerned, the text is intended for a rather sophisticated reader with at least some rudimentary knowledge of subjects like topology, functional analysis, and complex variables.

 

For this new edition the entire text has been carefully revised, and some portions are totally rewritten. More importantly, I have inserted more than a hundred pages of new material, in chapters on general measure and ergodic theory, the asymptotics of Markov processes, and large deviations. The expanded size has made it possible to give a self‑contained treatment of the underlying measure theory and to include topics like multivariate and ratio ergodic theorems, shift coupling, Palm distributions, entropy and information, Harris recurrence, invariant measures, strong and weak ergodicity, Strassen's law of the iterated logarithm, and the basic large deviation results of Cramer, Sanov, Schilder, and Reidlin and Ventzel.

Unfortunately, the body of knowledge in probability theory keeps growing at an ever-increasing rate, and I am painfully aware that I will never catch up in my efforts to survey the entire subject. Many areas are still totally beyond reach, and a comprehensive treatment of the more recent developments would require another volume or two.

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