Mathematics Books

Book SynopsisThis truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.Trade Review'Hacking does not restrict himself to the foundations of mathematics, but dares to cover both the breadth and the depth of mathematical philosophy.' Literary Review of Canada'… readable, presented in easily digestible chunks, clearly explained, and just a lot of fun …' Danny Yee's Book Reviews'Show[s] non-specialists … the sort of distinctive contribution to science and maths that a brilliant, very well-informed, philosopher can bring … I thoroughly recommend this book.' Alan Weir, The Times Literary Supplement'Hacking has composed a great overview of our understanding of mathematics and of the historical turning points and philosophical basics.' Peeter Müürsepp, Mathematical ReviewsTable of ContentsForeword; 1. A Cartesian introduction; 2. What makes mathematics mathematics?; 3. Why is there philosophy of mathematics?; 4. Proofs; 5. Applications; 6. In Plato's name; 7. Counter-Platonisms; Disclosures.

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Book SynopsisCovers key topics on the mathematics higher level for the IB diploma, offering test taking strategies and detailed answer solutions.Table of ContentsIntroduction; Additional advice; 1. Counting principles; 2. Exponentials and logs; 3. Polynomials; 4. Functions; 5. Sequences and series; 6. Trigonometry; 7. Vectors; 8. Complex numbers; 9. Differentiation; 10. Integration; 11. Probability and statistics; 12. Induction; Answers; Worked solutions.

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Book SynopsisCambridge International AS & A Level Further Mathematics supports students following the 9231 syllabus. This single coursebook comprehensively covers all four modules of the syllabus and helps support students in their studies and develops their mathematical skills. Authored by experienced teachers of Further Mathematics, the coursebook provides detailed explanations and clear worked examples with practice exercises and exam-style questions. Answers are at the back of the book.Table of ContentsIntroduction; Further Pure Mathematics 1: 1. Roots of polynomial equations; 2. Rational functions; 3. Summation of series; 4. Matrices 1; 5. Polar coordinates; 6. Vectors; 7. Proof by induction; Cross-topic revision exercise 1; Further Statistics: 8. Continuous random variables; 9. Inferential statistics; 10. Chi-squared tests; 11. Non-parametric tests; 12. Probability generating functions; Cross-topic revision exercise 2; Further Mechanics: 13. Projectiles; 14. Equilibrium of a rigid body; 15. Circular motion; 16. Hooke's law; 17. Linear motion under a variable force; 18. Momentum; Cross-topic revision exercise 3; Further Pure Mathematics 2: 19. Hyperbolic functions; 20. Matrices 2; 21. Differentiation; 22. Integration; 23. Complex numbers; 24. Differential equations; Cross-topic revision exercise 4; Practice paper; Glossary; Answers.

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Book SynopsisInteger linear programming (ILP) is a versatile modeling and optimization technique that is increasingly used in non-traditional ways in biology, with the potential to transform biological computation. However, few biologists know about it. This how-to and why-do text introduces ILP through the lens of computational and systems biology. It uses in-depth examples from genomics, phylogenetics, RNA, protein folding, network analysis, cancer, ecology, co-evolution, DNA sequencing, sequence analysis, pedigree and sibling inference, haplotyping, and more, to establish the power of ILP. This book aims to teach the logic of modeling and solving problems with ILP, and to teach the practical ''work flow'' involved in using ILP in biology. Written for a wide audience, with no biological or computational prerequisites, this book is appropriate for entry-level and advanced courses aimed at biological and computational students, and as a source for specialists. Numerous exercises and accompanying soTrade Review'In his classic accessible teaching style, Gusfield teaches us why integer linear programming (ILP) is the most useful mathematical idea you've probably never heard of. Read this book to learn how what you don't know can hurt you, and why ILP should be your new favorite method.' Trey Ideker, University of California, San Diego'Once again, Dan Gusfield has written an accessible book that shows that algorithmic rigor need not be sacrificed when solving real-world problems. He explains integer linear programming in the context of real-world biology. In doing so, the reader has an enriched understanding of both algorithmic details and the challenges in modern biology.' Russ Altman, Stanford University, CaliforniaTable of ContentsPreface; Part I: 1. A fly-over introduction; 2. Biological networks and graphs; 3. Character compatibility; 4. Near-cliques; 5. Parsimony in phylogenetics; 6. RNA folding; 7. Protein problems; 8. Tanglegrams; 9. TSP in genomics; 10. Molecular sequence analysis; 11. Metabolic networks and engineering; 12. ILP idioms; Part II: 13. Communities and cuts; 14. Corrupted data and extensions in phylogenetics; 15. More tanglegrams and trees; 16. Return to Steiner-trees; 17. Exploiting protein networks; 18. More strings and sequences; 19. Max-likelihood pedigrees; 20. Haplotyping; 21. Extended exercises; 22. What's next?; Epilogue: opinionated comments.

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Quadratics; 2. Functions; 3. Coordinate geometry; 4. Circular measure; 5. Trigonometry; 6. Series; 7. Differentiation; 8. Further differentiation; 9. Integration; Answers

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Representation of data; 2. Measures of central tendency; 3. Measures of variation; 4. Probability; 5. Permutations and combinations; 6. Probability distributions; 7. The binomial and geometric distributions; 8. The normal distribution; Answers

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Hypothesis testing; 2. The Poisson distribution; 3. Linear combinations of random variables; 4. Continuous random variables; 5. Sampling; 6. Estimation; Answers

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Book SynopsisFor Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing ''aspects'', or dimensions of meaning, projecting mathematical ''realities''. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein''s Later Philosophy of Mathematics (19341951) grew from his Early (19121921) and Middle (192933) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.Trade Review'Brimming with ideas from, and analyses of, Wittgenstein's philosophy of mathematics, the book contains both material of interest for those new to the topic, as well as often subtle and novel interpretations aimed at experts … A rich and sophisticated take on an intricate topic, Floyd's book is to be warmly recommended to all those eager to deepen their understanding of Wittgenstein's philosophy of mathematics.' Sorin Bangu, Philosophia MathematicaTable of Contents1. Introduction; 2. Early Philosophy: Absolute Simplicity; 3. Middle Philosophy (1929–1933): Relative Simplicity; 4. Later Philosophy: Fluid Simplicity (1937–1951).

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020.Table of ContentsIntroduction; 1. Velocity and acceleration; 2. Force and motion in one dimension; 3. Forces in two dimensions; 4. Friction; 5. Connected particles; 6. General motion in a straight line; 7. Momentum; 8. Work and energy; 9. The work–energy principle and power; Answers

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Book SynopsisThis series has been developed specifically for the Cambridge International AS & A Level Mathematics (9709) syllabus to be examined from 2020. The coursebook provides clear explanations and practice exercises to help students master maths skills. This edition comes with a subscription to the Probability & Statistics 1 Cambridge Online Mathematics component. With all of the materials found within the coursebook, including answers, Cambridge Online Mathematics offers students the facility to demonstrate their working, as well as opportunities for self-assessment, and allows teachers to set classroom and homework exercises to individual students or classes, with the ability to track progress. The online resource requires internet access. For more information on how to use Cambridge Online Mathematics, please see inside the front cover.

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Book SynopsisWritten by a world expert on the subject, Origametry is the first complete reference on the mathematics of origami. It is an essential reference for researchers of origami mathematics and applications in physics, engineering, and design. Educators, students, and enthusiasts will also enjoy this fascinating account of the mathematics of folding.Trade Review'This is a magnificent, comprehensive work. It gives a thorough overview of the underlying mathematics of flat-facet (polyhedral) origami, bringing together diverse contributions from many investigators (including the author's own seminal work), along with historical notes and context that ties everything together. This will be the standard reference for the mathematics of origami for years to come, and with the plethora of open problems, will also undoubtedly be the inspiration for many master's and Ph.D. theses in the future!' Robert J. Lang, author of Origami Design Secrets and Twists, Tilings, and Tessellations'Tom Hull has always been the authority and historian on origami mathematics. In this beautiful book, he ties together a wide range of classic and modern results, grounding them in their rich history.' Erik Demaine, Massachusetts Institute of Technology'Fans of classical geometry, geometry or topology of a combinatorial flavor, and even algebra and analysis will find much to appreciate in this book. Students and instructors alike will discover open problems, elegant theorems, and historical digressions, along with fascinating applications of concepts from standard courses. Origametry could easily function as a sourcebook for further explorations, capstone projects, or original research.' D. P. Tumer, Notices of the American Mathematical Society'… a delightful and informative read for mathematicians curious about the mathematics behind origami, essential for researchers starting out in this area, and handy for educators searching for ideas in topics connecting mathematics, origami and its applications.' Ana Rita Pires, European Mathematical Society MagazineTable of ContentsIntroduction; Part I. Geometric Constructions: 1. Examples and basic folds; 2. Solving equations via folding; 3. Origami algebra; 4. Beyond classic origami; Part II. The Combinatorial Geometry of Flat Origami: 5. Flat vertex folds: local properties; 6. Multiple-vertex flat folds: global properties; 7. Counting flat folds; 8. Other flat folding problems; Part III. Algebra, Topology, and Analysis in Origami: 9. Origami homomorphisms; 10. Folding manifolds; 11. An analytic approach to isometric foldings; Part IV. Non-Flat Folding: 12. Rigid origami; 13. Rigid foldings; 14. Rigid origami theory; References; Index.

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Book SynopsisWritten by an experienced author and teacher, this print book with Cambridge Elevate edition provides fully-worked solutions for exercises contained within the Cambridge International AS & A Level Mathematics Pure Mathematics 2 & 3 Coursebook. This manual is designed to help save you time with over 1,000 readily-available solutions. We provide a selection of solutions to the questions in the coursebook to enable the learners to develop as confident, independent thinkers. The solutions to our practice exam-style paper, end-of-chapter review exercises and cross-topic review exercises can be found on the digital part of the resource.

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Book SynopsisWritten by an experienced author and teacher team, this print book with Cambridge Elevate edition provides fully-worked solutions for exercises contained within the Cambridge International AS & A Level Mathematics Mechanics Coursebook. This manual is designed to help save you time with over 1,000 readily-available solutions. The solutions to our practice exam-style paper, end-of-chapter review exercises and cross-topic review exercises can be found on the digital part of the resource.

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Book SynopsisWritten by an experienced author and teacher team, this print book with Cambridge Elevate edition provides fully-worked solutions for exercises contained within the Cambridge International AS & A Level Further Mathematics Coursebook. This manual is designed to help save you time with over 1,000 readily-available solutions. The solutions to our practice exam papers, end of chapter and cross-topic reviews can be found on the digital part of the resource.

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Book SynopsisThis elegant introduction to symplectic geometry and Lagrangian foliations is suitable for students at both the undergraduate and the graduate level. It includes both new expositions of standard material and new material that has not appeared in book form previously, and provides a systematic study of bi-Lagrangian geometry.

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Book SynopsisDeep learning on graphs has become one of the hottest topics in machine learning. The book consists of four parts to best accommodate our readers with diverse backgrounds and purposes of reading. Part 1 introduces basic concepts of graphs and deep learning; Part 2 discusses the most established methods from the basic to advanced settings; Part 3 presents the most typical applications including natural language processing, computer vision, data mining, biochemistry and healthcare; and Part 4 describes advances of methods and applications that tend to be important and promising for future research. The book is self-contained, making it accessible to a broader range of readers including (1) senior undergraduate and graduate students; (2) practitioners and project managers who want to adopt graph neural networks into their products and platforms; and (3) researchers without a computer science background who want to use graph neural networks to advance their disciplines.Trade Review'This timely book covers a combination of two active research areas in AI: deep learning and graphs. It serves the pressing need for researchers, practitioners, and students to learn these concepts and algorithms, and apply them in solving real-world problems. Both authors are world-leading experts in this emerging area.' Huan Liu, Arizona State University'Deep learning on graphs is an emerging and important area of research. This book by Yao Ma and Jiliang Tang covers not only the foundations, but also the frontiers and applications of graph deep learning. This is a must-read for anyone considering diving into this fascinating area.' Shuiwang Ji, Texas A&M University'The first textbook of Deep Learning on Graphs, with systematic, comprehensive and up-to-date coverage of graph neural networks, autoencoder on graphs, and their applications in natural language processing, computer vision, data mining, biochemistry and healthcare. A valuable book for anyone to learn this hot theme!' Jiawei Han, University of Illinois at Urbana-Champaign'This book systematically covers the foundations, methodologies, and applications of deep learning on graphs. Especially, it comprehensively introduces graph neural networks and their recent advances. This book is self-contained and nicely structured and thus suitable for readers with different purposes. I highly recommend those who want to conduct research in this area or deploy graph deep learning techniques in practice to read this book.' Charu Aggarwal, Distinguished Research Staff Member at IBM and recipient of the W. Wallace McDowell AwardTable of Contents1. Deep Learning on Graphs: An Introduction; 2. Foundation of Graphs; 3. Foundation of Deep Learning; 4. Graph Embedding; 5. Graph Neural Networks; 6. Robust Graph Neural Networks; 7. Scalable Graph Neural Networks; 8. Graph Neural Networks for Complex Graphs; 9. Beyond GNNs: More Deep Models for Graphs; 10. Graph Neural Networks in Natural Language Processing; 11. Graph Neural Networks in Computer Vision; 12. Graph Neural Networks in Data Mining; 13. Graph Neural Networks in Biochemistry and Healthcare; 14. Advanced Topics in Graph Neural Networks; 15. Advanced Applications in Graph Neural Networks.

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Book SynopsisThis upper undergraduate abstract algebra text covers classical themes on groups, rings and fields in depth, augmented with a strong emphasis on irreducible polynomials, a fresh approach to modules and linear algebra, a fresh take on Gröbner theory, and a group theoretic treatment of Rejewski's deciphering of the Enigma machine.Trade Review'This is a very good book, which provides an excellent introduction to modern algebra for senior undergraduate or beginning graduate students. The book includes a thorough coverage of the standard topics in the theories of groups, rings, fields, modules and Galois theory, taking a conceptual approach to algebra. For instance, the group theory part focuses on group actions, the ring theory exposition very appropriately stresses unique factorization properties, and the Galois theory part details some rather conceptual applications. Some of the less standard, very interesting topics are also present, including the breaking of the Enigma machine, as well as an in-depth look at division algorithms, including Gröbner bases. The book includes numerous exercises. All in all, a great new algebra text!' Lenny Fukshansky, Claremont McKenna College'An excellent textbook for an advanced undergraduate or a beginning graduate course on abstract algebra. Includes a lucid discussion of all core topics in group theory, commutative ring theory, Galois theory, and modules over principal ideal domains. I would describe this book as a simplified version of the classical textbook by Dummit and Foote.' Mihran Papikian, Pennsylvania State University'The 'comprehensive' in the title is no joke: this book walks the reader through a sea of detailed examples and computations in abstract algebra. These, and the exercises, are well thought out and will appeal to the student who likes a very hands-on kind of textbook. The group actions and division algorithms chapters are my personal favourites, as the computational nature of those topics plays to the strengths of the authors.' Nick Gurski, Case Western Reserve University'This is a great introduction to abstract algebra for graduate students and mathematically mature undergraduates.' Thomas Garrity, Williams College'Lawrence and Zorzitto's treatment of Abstract Algebra is lucid and thorough. I am particularly pleased to see the inclusion of Gröbner basis theory in a way that is accessible to introductory students, as it makes possible the exploration of polynomial ideals to great depth.' Jeffrey Clark, Elon UniversityTable of ContentsContents; Preface; 1. A refresher on the integers; 2. A first look at groups; 3. Groups acting on sets; 4. Basics on rings-mostly commutative; 5. Primes and unique factorization; 6. Algebraic field extensions; 7. Applications of galois theory; 8. Modules over principal ideal domains; 9. Division algorithms; Appendix A: Infinite sets.

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Book SynopsisPaul Erdos published more papers during his lifetime than any other mathematician, especially in discrete mathematics. He had a nose for beautiful, simply-stated problems with solutions that have far-reaching consequences across mathematics. This captivating book, written for students, provides an easy-to-understand introduction to discrete mathematics by presenting questions that intrigued Erdos, along with his brilliant ways of working toward their answers. It includes young Erdos''s proof of Bertrand''s postulate, the Erdos-Szekeres Happy End Theorem, De Bruijn-Erdos theorem, Erdos-Rado delta-systems, Erdos-Ko-Rado theorem, Erdos-Stone theorem, the Erdos-Rényi-Sós Friendship Theorem, Erdos-Rényi random graphs, the Chvátal-Erdos theorem on Hamilton cycles, and other results of Erdos, as well as results related to his work, such as Ramsey''s theorem or Deza''s theorem on weak delta-systems. Its appendix covers topics normally missing from introductory courses. Filled with personal aneTrade Review'Vašek Chvátal was born to write this one-of-a-kind book. Readers cannot help but be captivated by the evident love with which every page has been written. The human side of mathematics is intertwined beautifully with first-rate exposition of first-rate results.' Donald Knuth, Stanford University'This book is a treasure trove from so many viewpoints. It is a wonderful introduction and an alluring invitation to discrete mathematics - now a central field of mathematics identified mostly with the hero of this book. With lucid, carefully planned chapters on different topics it demonstrates the unique way in which Paul Erdős, one of the most prolific and influential mathematicians of the twentieth century, invented and approached problems. Sprinkled with historical and personal anecdotes and pictures, it opens a window to the unique personality of 'Uncle Paul'. And implicitly, it reveals the charming and candid way in which Vašek Chvátal, an authority in the field and a lifelong friend and collaborator of Erdős, likes to combine teaching and story-telling.' Avi Wigderson, IAS, Princeton'Paul Erdős is one of the founding fathers of modern combinatorics, whose ability to pose beautiful problems greatly determined the development of this field and influenced many other areas of mathematics. This book uses some basic questions, which intrigued Paul Erdős, to give a nice introduction to many topics in discrete mathematics. It contains a collection of beautiful results, covering such diverse subjects as discrete geometry, Ramsey theory, graph colorings, extremal problems for graphs and set systems and some others. It presents many elegant proofs and exposes the reader to various powerful combinatorial techniques.' Benjamin Sudakov, ETH Zurich'This is a brilliant book. It manages in one fell swoop to survey and develop a large part of combinatorial mathematics while at the same time chronicling the work of Paul Erdős. His contributions to different areas of mathematics are seen here to be part of a coherent whole. Chvátal's presentation is particularly appealing and accessible. The wonderful personal recollections add to the mathematical content to provide a portrait of Erdős' mind recognizable to those who knew him.' Bruce Rothschild, University of California, Los Angeles'Vašek Chvátal's book is a gem. Paul Erdős' favorite problems and best work are beautifully laid out. Readers unfamiliar with Erdős' work cannot fail to appreciate its power and elegance, and those who have seen bits and pieces will have the pleasure of seeing it thoughtfully and lovingly presented by a master. It's hard to imagine now, but there was a time when combinatorics was thought to be a jumble of results without depth or coherence. 'Uncle' Paul understood its heart and soul, and nowhere is this more evident than in Chvátal's wonderful compendium. This volume belongs on every math-lover's night-table!' Peter Winkler, Dartmouth College'Beautiful mathematics is presented with great care and clarity in Vašek Chvátal's book, complemented with well-written anecdotes and personal reminiscences about Paul Erdős. This combination makes the book a very enjoyable reading and a lively tribute to the memory of one of the most prolific mathematicians of all time. Studying discrete mathematics from this book is likely to give a great experience to students and established researchers alike.' Gábor Simonyi, Rényi Institute, Budapest'… Chvátal (emer., Concordia Univ.) has created a gem in this work and deserves congratulation … Highly recommended.' J. Johnson, Choice Magazine'This wonderfully written book is undoubtedly a significant contribution to the growing body of literature on the various developments in discrete mathematics over the last several decades. Still, to reduce it to only its mathematical dimension would be an act of injustice not only towards the book but also towards its author. The book is a powerful homage to Paul Erdos as one of the leading mathematicians of the twentieth century as well as a person who, with his unprecedented level of academic generosity and overall human kindness, was one of the pillars of the discrete mathematics community during his lifetime.' Veselin Jungic, MathSciNetTable of ContentsForeword; Preface; Acknowledgments; Introduction; 1. A glorious beginning – Bertrand's postulate; 2. Discrete geometry and spinoffs; 3. Ramsey's theorem; 4. Delta-systems; 5. Extremal set theory; 6. Van der Waerden's theorem; 7. Extremal graph theory; 8. The friendship theorem; 9. Chromatic number; 10. Thresholds of graph properties ; 11. Hamilton cycles; Appendix A. A few tricks of the trade; Appendix B. Definitions, terminology, notation; Appendix C. More on Erdős; References; Index.

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Book SynopsisNumerical relativity has emerged as the key tool to model gravitational waves - recently detected for the first time - that are emitted when black holes or neutron stars collide. This book provides a pedagogical, accessible, and concise introduction to the subject. Relying heavily on analogies with Newtonian gravity, scalar fields and electromagnetic fields, it introduces key concepts of numerical relativity in a context familiar to readers without prior expertise in general relativity. Readers can explore these concepts by working through numerous exercises, and can see them ''in action'' by experimenting with the accompanying Python sample codes, and so develop familiarity with many techniques commonly employed by publicly available numerical relativity codes. This is an attractive, student-friendly resource for short courses on numerical relativity, as well as providing supplementary reading for courses on general relativity and computational physics.Trade Review'Computational general relativity has now become a central tool for the exploration of the astrophysical universe, and gravitational-wave astronomy would not be possible without it. A burgeoning or seasoned astrophysicist who wishes to be up to date must therefore acquire an awareness of the field's methods and main achievements. But where to begin? With this book! Baumgarte and Shapiro are leading experts (indeed, founding experts) of this field, and with their trademark lucid and engaging prose, they take us gently by the hand on a comprehensive guided tour. Mysterious notions (lapse, shift, extrinsic curvature, constraint equations) are introduced seamlessly, and the book features a gallery of the field's most important results to date. A superb achievement for the great benefit of the scientific community.' Eric Poisson, University of Guelph; author of A Relativist's Toolkit'Numerical relativity well deserves its reputation as a subject of great beauty yet prodigious conceptual difficulty and daunting technical complexity. This outstanding text, by two leading practitioners of the field, is a wonderful Rosetta Stone for those seeking an efficient path toward a working knowledge of the subject. For me it will serve as an essential reference. I'm sorry only that it was not available sooner.' Robert Eisenstein, Massachusetts Institute of Technology'This is an excellent book explaining the general relativistic two-body problem and its numerical treatment in a highly pedagogical manner to a broad scientific audience. Besides the main topic, readers will also gain some unexpected insight and new viewpoints on numerous wider aspects of Einstein's theory.' Ulrich Sperhake, University of Cambridge'Black holes and gravitational waves are, thanks to new observations, fast-advancing frontiers of astronomy that attract wide interest. Their implications are best addressed by powerful computers, so this text, by two acknowledged world experts, is especially welcome and timely.' Martin Rees, Astronomer Royal; author of Gravity's Fatal AttractionTable of ContentsPreface; 1. Newton's and Einstein's gravity; 2. Foliations of spacetime: constraint and evolution equations; 3. Solving the constraint equations; 4 Solving the evolution equations; 5. Numerical simulations of black-hole binaries; Epilogue; Appendix A. A brief review of tensor properties; Appendix B. A brief introduction to some numerical techniques; Appendix C. A very brief introduction to matter sources; Appendix D. A summary of important results; Appendix E. Answers to selected problems; References; Index.

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Book SynopsisUsing a unique pedagogical approach, this text introduces mathematical logic by guiding students in implementing the underlying logical concepts and mathematical proofs via Python programming. This approach, tailored to the unique intuitions and strengths of the ever-growing population of programming-savvy students, brings mathematical logic into the comfort zone of these students and provides clarity that can only be achieved by a deep hands-on understanding and the satisfaction of having created working code. While the approach is unique, the text follows the same set of topics typically covered in a one-semester undergraduate course, including propositional logic and first-order predicate logic, culminating in a proof of Gödel''s completeness theorem. A sneak peek to Gödel''s incompleteness theorem is also provided. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. Familiarity with proofs and basic proficiency in Python is assumed.Trade Review'The authors transformed the first course in Mathematical Logic – an experience that many students view as daunting and technical – into an inspiring journey that sails playfully yet rigorously from logic's first principles to Gödel's Completeness Theorem. The secret sauce is making progress by writing many little Python programs instead of proving theorems, a hands-on approach that suits computer science students perfectly.' Shimon Schocken, Reichman University'Mathematical logic is all about expressions and syntactic operations, and many of its best ideas find a natural home in computer science. Gonczarowski and Nisan make the subject come alive by opening it up to computational implementation and exploration.' Jeremy Avigad, Carnegie Mellon University'Mathematical Logic through Python offers a refreshingly innovative approach that makes it stand out among several excellent books on mathematical logic. By building on readers' experience and intuition through programming, it naturally provides them with a deep understanding of the fundamental concepts of mathematical logic that underly computer science.' Yoram Moses, Technion - Israel Institute of TechnologyTable of ContentsPreface; Introduction and Overview; Part I. Propositional Logic: 1. Propositional Logic Syntax; 2. Propositional Logic Semantics; 3. Logical Operators; 4. Proof by Deduction; 5. Working with Proofs; 6. The Tautology Theorem and the Completeness of Propositional Logic; Part II. Predicate Logic: 7. Predicate Logic Syntax and Semantics; 8. Getting Rid of Functions and Equality; 9. Deductive Proofs of Predicate Logic Formulas; 10. Working with Predicate Logic Proofs; 11. The Deduction Theorem and Prenex Normal Form; 12. The Completeness Theorem; 13. Sneak Peek at Mathematical Logic II: Godel's Incompleteness Theorem; Cheatsheet Axioms and Axiomatic Inference Rules Used in this Book; Notes; Index.

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Book SynopsisThe study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors'' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painlevé equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.Trade Review'… a stylish, well-written and up to date introduction to complex variable methods for undergraduate (or early graduate) students in applied mathematics, science and engineering … I thoroughly enjoyed reading this book and warmly commend it to anyone seeking a brisk, well-organised account of complex variables with a practical focus on applications and calculational aspects.' Nick Lord, The Mathematical GazetteTable of Contents1. Complex numbers and elementary functions; 2. Analytic functions and integration; 3. Sequences, series and singularities of complex functions; 4. Residue calculus and applications of contour integration; 5. Conformal mappings and applications; Appendix. Answers to selected odd-numbered exercises; References; Index.

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Book SynopsisDo you want to know what a parametric test is and when not to perform one? Do you get confused between odds ratios and relative risks? Want to understand the difference between sensitivity and specificity? Would like to find out what the fuss is about Bayes'' theorem? Then this book is for you! Physicians need to understand the principles behind medical statistics. They don''t need to learn the formula. The software knows it already! This book explains the fundamental concepts of medical statistics so that the learner will become confident in performing the most commonly used statistical tests. Each chapter is rich in anecdotes, illustrations, questions, and answers. Not enough? There is more material online with links to free statistical software, webpages, multimedia content, a practice dataset to get hands-on with data analysis, and a Single Best Answer questionnaire for the exam.Trade Review'An accessible book by a practising doctor, aimed at other doctors, which explains key statistical concepts in words and pictures. An excellent foundation for those seeking to understand the numbers in medical journal articles and quantitative reports.' Professor Trish Greenhalgh, University of Oxford, UK'Statistics forms the starting point for evidence based medicine, though most medics would argue that their own statistical awareness is still near the starting point! This book eases you into the awesome, exciting, exhilarating world of statistics, and makes you understand just how cool it really is. It will unleash your inner statistician that no-one knew existed - especially you!' Professor Dan Perry, Children's Orthopaedic Surgeon and Fellow of Wolfson College, University of Oxford, UK'The book provides a light-hearted introduction to the basic concepts in medical statistics. A couple of hundred pages long with short chapters, the book delivers with clear focus the key statistical concepts alongside some general knowledge to lighten what is sometimes a very arid subject. The description of concepts with graphs and figures support the visual learner. I thoroughly enjoyed the quick questions presented alongside the description of concepts to test understanding, with the answers at the end of chapter which linked to bullet point summaries, help to consolidate the concepts covered. I thought it was an excellent way for someone to start on their path to understanding this area. Finally, I particularly appreciated the last chapter with its focus on the work by our dear friend Doug Altman.' Professor Rafael Perera, Professor of Medical Statistics, University of Oxford, UK'This is an excellent introductory book for medical statistics. It's well written, easy to read, with some great examples of statistics in everyday clinical practice. The question and answer format is especially useful in reinforcing key concepts discussed in the chapter. There are lots of additional learning material included in the online resource for those seeking a more detailed understanding of the topic. The author is to be congratulated on making an important but difficult subject appear relatively straightforward and interesting to even the non-expert.' Professor Paul Banaszkiewicz, Consultant Orthopaedic Surgeon North East NHS Surgical Centre (NENSC), Gateshead, UK, and Visiting Professor Northumbria University, Newcastle-upon-Tyne, UK 'The modern practice in orthopaedic and trauma surgery is a completely different practice to that which many of us grew up with and is now firmly founded on research and evidence. And this evidence itself is built around scientific method and statistical analysis. This excellent book provides a comprehensive guide to biostatistics for the orthopaedic surgeon and aspiring clinician scientist. Through clear explanations of complex concepts the author succeeds in simplifying the difficult and I am sure this will become an essential reference for all involved in orthopaedic surgery.' Professor Ben Ollivere, Professor of Orthopaedic Trauma, University of Nottingham, UKTable of ContentsPreface; Acknowledgements; How to get the best out of this book; 1. Medicine and numbers: what is the connection?; 2. Measuring a variable: what is the difference between eye colour and height; 3. Summarising data: communicating easily; 4. Why average and range is not always enough: standard deviation and standard error; 5. The normal distribution: what's so 'normal' about it?; 6. Confidence interval: what is your guesstimate?; 7. Innocent until proven guilty! The null hypothesis; 8. Errors in hypothesis tests: learn your α from your β; 9. The randomized controlled trial: why does it have to be random?; 10. Choosing a statistical test: to t or not to t?; 11. Finding the odd one out: the ANOVA test; 12. Categorically different? The Chi-Squared test; 13. If the line fits: correlation and linear regression; 14. Hindsight is 20/20: logistic regression; 15. Don't risk the odds. risk vs odds as the outcome measure; 16. I will survive! Time to event data analysis; 17. High-ceiling or low threshold? Accuracy of a diagnostic test; 18. Apples or oranges? Meta-analysis of selected studies; 19. Lies, damned lies and statistics: untangling facts from fiction!; Glossary; Appendix 1. Are you ready to test yourself? Single Best Answer questionnaire; Appendix 2. Software and practice problems; Appendix 3. Practice database; Appendix 4. References and further reading; Appendix 5. Further resources.

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Book SynopsisThis text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student''s level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofsalso explains when each technique is likely to be used, based on certain key words that appear in the problem under consideration. Doing so enables students to choose a technique consciously, based on the form of the problem.Trade Review"The instructional material is to the point, with well-considered examples and asides on common mistakes. Good examples of the author's thoughtfulness appear in the discourses on pp. 5-6 of identifying the hypothesis and conclusion when they are not obvious, on pp. 28-29 regarding overlapping notation, and on pp. 190-191 of the advantages and disadvantages of generalization." (Zentralblatt MATH 2016)Table of ContentsForeword xi Preface to the Student xiii Preface to the Instructor xv Acknowledgments xviii Part I Proofs 1 Chapter 1: The Truth of It All 1 2 The Forward-Backward Method 9 3 On Definitions and Mathematical Terminology 25 4 Quantifiers I: The Construction Method 41 5 Quantifiers II: The Choose Method 53 6 Quantifiers III: Specialization 69 7 Quantifiers IV: Nested Quantifiers 81 8 Nots of Nots Lead to Knots 93 9 The Contradiction Method 101 10 The Contrapositive Method 115 11 The Uniqueness Methods 125 12 Induction 133 13 The Either/Or Methods 145 14 The Max/Min Methods 155 15 Summary 163 Part II Other Mathematical Thinking Processes 16 Generalization 179 17 Creating Mathematical Definitions 197 18 Axiomatic Systems 219 Appendix A Examples of Proofs from Discrete Mathematics 237 Appendix B Examples of Proofs from Linear Algebra 251 Appendix C Examples of Proofs from Modern Algebra 269 Appendix D Examples of Proofs from Real Analysis 287 Solutions to Selected Exercises 305 Glossary 357 References 367 Index 369

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Book SynopsisBayesian Statistics is the school of thought that combines prior beliefs with the likelihood of a hypothesis to arrive at posterior beliefs. The first edition of Peter Lee s book appeared in 1989, but the subject has moved ever onwards, with increasing emphasis on Monte Carlo based techniques.Trade Review“As a lifelong non-statistician and sporadic “user” of statistics, I have not come across another advanced statistics book (as I would characterize this one) that offers so much to the non-expert and, I’ll bet, to the expert as well. The book has my highest recommendation.” (Computing Reviews, 7 January 2013)Table of ContentsPreface xix Preface to the First Edition xxi 1 Preliminaries 1 1.1 Probability and Bayes’ Theorem 1 1.1.1 Notation 1 1.1.2 Axioms for probability 2 1.1.3 ‘Unconditional’ probability 5 1.1.4 Odds 6 1.1.5 Independence 7 1.1.6 Some simple consequences of the axioms; Bayes’ Theorem 7 1.2 Examples on Bayes’ Theorem 9 1.2.1 The Biology of Twins 9 1.2.2 A political example 10 1.2.3 A warning 10 1.3 Random variables 12 1.3.1 Discrete random variables 12 1.3.2 The binomial distribution 13 1.3.3 Continuous random variables 14 1.3.4 The normal distribution 16 1.3.5 Mixed random variables 17 1.4 Several random variables 17 1.4.1 Two discrete random variables 17 1.4.2 Two continuous random variables 18 1.4.3 Bayes’ Theorem for random variables 20 1.4.4 Example 21 1.4.5 One discrete variable and one continuous variable 21 1.4.6 Independent random variables 22 1.5 Means and variances 23 1.5.1 Expectations 23 1.5.2 The expectation of a sum and of a product 24 1.5.3 Variance, precision and standard deviation 25 1.5.4 Examples 25 1.5.5 Variance of a sum; covariance and correlation 27 1.5.6 Approximations to the mean and variance of a function of a random variable 28 1.5.7 Conditional expectations and variances 29 1.5.8 Medians and modes 31 1.6 Exercises on Chapter 1 31 2 Bayesian inference for the normal distribution 36 2.1 Nature of Bayesian inference 36 2.1.1 Preliminary remarks 36 2.1.2 Post is prior times likelihood 36 2.1.3 Likelihood can be multiplied by any constant 38 2.1.4 Sequential use of Bayes’ Theorem 38 2.1.5 The predictive distribution 39 2.1.6 A warning 39 2.2 Normal prior and likelihood 40 2.2.1 Posterior from a normal prior and likelihood 40 2.2.2 Example 42 2.2.3 Predictive distribution 43 2.2.4 The nature of the assumptions made 44 2.3 Several normal observations with a normal prior 44 2.3.1 Posterior distribution 44 2.3.2 Example 46 2.3.3 Predictive distribution 47 2.3.4 Robustness 47 2.4 Dominant likelihoods 48 2.4.1 Improper priors 48 2.4.2 Approximation of proper priors by improper priors 49 2.5 Locally uniform priors 50 2.5.1 Bayes’ postulate 50 2.5.2 Data translated likelihoods 52 2.5.3 Transformation of unknown parameters 52 2.6 Highest density regions 54 2.6.1 Need for summaries of posterior information 54 2.6.2 Relation to classical statistics 55 2.7 Normal variance 55 2.7.1 A suitable prior for the normal variance 55 2.7.2 Reference prior for the normal variance 58 2.8 HDRs for the normal variance 59 2.8.1 What distribution should we be considering? 59 2.8.2 Example 59 2.9 The role of sufficiency 60 2.9.1 Definition of sufficiency 60 2.9.2 Neyman’s factorization theorem 61 2.9.3 Sufficiency principle 63 2.9.4 Examples 63 2.9.5 Order statistics and minimal sufficient statistics 65 2.9.6 Examples on minimal sufficiency 66 2.10 Conjugate prior distributions 67 2.10.1 Definition and difficulties 67 2.10.2 Examples 68 2.10.3 Mixtures of conjugate densities 69 2.10.4 Is your prior really conjugate? 71 2.11 The exponential family 71 2.11.1 Definition 71 2.11.2 Examples 72 2.11.3 Conjugate densities 72 2.11.4 Two-parameter exponential family 73 2.12 Normal mean and variance both unknown 73 2.12.1 Formulation of the problem 73 2.12.2 Marginal distribution of the mean 75 2.12.3 Example of the posterior density for the mean 76 2.12.4 Marginal distribution of the variance 77 2.12.5 Example of the posterior density of the variance 77 2.12.6 Conditional density of the mean for given variance 77 2.13 Conjugate joint prior for the normal distribution 78 2.13.1 The form of the conjugate prior 78 2.13.2 Derivation of the posterior 80 2.13.3 Example 81 2.13.4 Concluding remarks 82 2.14 Exercises on Chapter 2 82 3 Some other common distributions 85 3.1 The binomial distribution 85 3.1.1 Conjugate prior 85 3.1.2 Odds and log-odds 88 3.1.3 Highest density regions 90 3.1.4 Example 91 3.1.5 Predictive distribution 92 3.2 Reference prior for the binomial likelihood 92 3.2.1 Bayes’ postulate 92 3.2.2 Haldane’s prior 93 3.2.3 The arc-sine distribution 94 3.2.4 Conclusion 95 3.3 Jeffreys’ rule 96 3.3.1 Fisher’s information 96 3.3.2 The information from several observations 97 3.3.3 Jeffreys’ prior 98 3.3.4 Examples 98 3.3.5 Warning 100 3.3.6 Several unknown parameters 100 3.3.7 Example 101 3.4 The Poisson distribution 102 3.4.1 Conjugate prior 102 3.4.2 Reference prior 103 3.4.3 Example 104 3.4.4 Predictive distribution 104 3.5 The uniform distribution 106 3.5.1 Preliminary definitions 106 3.5.2 Uniform distribution with a fixed lower endpoint 107 3.5.3 The general uniform distribution 108 3.5.4 Examples 110 3.6 Reference prior for the uniform distribution 110 3.6.1 Lower limit of the interval fixed 110 3.6.2 Example 111 3.6.3 Both limits unknown 111 3.7 The tramcar problem 113 3.7.1 The discrete uniform distribution 113 3.8 The first digit problem; invariant priors 114 3.8.1 A prior in search of an explanation 114 3.8.2 The problem 114 3.8.3 A solution 115 3.8.4 Haar priors 117 3.9 The circular normal distribution 117 3.9.1 Distributions on the circle 117 3.9.2 Example 119 3.9.3 Construction of an HDR by numerical integration 120 3.9.4 Remarks 122 3.10 Approximations based on the likelihood 122 3.10.1 Maximum likelihood 122 3.10.2 Iterative methods 123 3.10.3 Approximation to the posterior density 123 3.10.4 Examples 124 3.10.5 Extension to more than one parameter 126 3.10.6 Example 127 3.11 Reference posterior distributions 128 3.11.1 The information provided by an experiment 128 3.11.2 Reference priors under asymptotic normality 130 3.11.3 Uniform distribution of unit length 131 3.11.4 Normal mean and variance 132 3.11.5 Technical complications 134 3.12 Exercises on Chapter 3 134 4 Hypothesis testing 138 4.1 Hypothesis testing 138 4.1.1 Introduction 138 4.1.2 Classical hypothesis testing 138 4.1.3 Difficulties with the classical approach 139 4.1.4 The Bayesian approach 140 4.1.5 Example 142 4.1.6 Comment 143 4.2 One-sided hypothesis tests 143 4.2.1 Definition 143 4.2.2 P-values 144 4.3 Lindley’s method 145 4.3.1 A compromise with classical statistics 145 4.3.2 Example 145 4.3.3 Discussion 146 4.4 Point (or sharp) null hypotheses with prior information 146 4.4.1 When are point null hypotheses reasonable? 146 4.4.2 A case of nearly constant likelihood 147 4.4.3 The Bayesian method for point null hypotheses 148 4.4.4 Sufficient statistics 149 4.5 Point null hypotheses for the normal distribution 150 4.5.1 Calculation of the Bayes’ factor 150 4.5.2 Numerical examples 151 4.5.3 Lindley’s paradox 152 4.5.4 A bound which does not depend on the prior distribution 154 4.5.5 The case of an unknown variance 155 4.6 The Doogian philosophy 157 4.6.1 Description of the method 157 4.6.2 Numerical example 157 4.7 Exercises on Chapter 4 158 5 Two-sample problems 162 5.1 Two-sample problems – both variances unknown 162 5.1.1 The problem of two normal samples 162 5.1.2 Paired comparisons 162 5.1.3 Example of a paired comparison problem 163 5.1.4 The case where both variances are known 163 5.1.5 Example 164 5.1.6 Non-trivial prior information 165 5.2 Variances unknown but equal 165 5.2.1 Solution using reference priors 165 5.2.2 Example 167 5.2.3 Non-trivial prior information 167 5.3 Variances unknown and unequal (Behrens–Fisher problem) 168 5.3.1 Formulation of the problem 168 5.3.2 Patil’s approximation 169 5.3.3 Example 170 5.3.4 Substantial prior information 170 5.4 The Behrens–Fisher controversy 171 5.4.1 The Behrens–Fisher problem from a classical standpoint 171 5.4.2 Example 172 5.4.3 The controversy 173 5.5 Inferences concerning a variance ratio 173 5.5.1 Statement of the problem 173 5.5.2 Derivation of the F distribution 174 5.5.3 Example 175 5.6 Comparison of two proportions; the 2 × 2 table 176 5.6.1 Methods based on the log-odds ratio 176 5.6.2 Example 177 5.6.3 The inverse root-sine transformation 178 5.6.4 Other methods 178 5.7 Exercises on Chapter 5 179 6 Correlation, regression and the analysis of variance 182 6.1 Theory of the correlation coefficient 182 6.1.1 Definitions 182 6.1.2 Approximate posterior distribution of the correlation coefficient 184 6.1.3 The hyperbolic tangent substitution 186 6.1.4 Reference prior 188 6.1.5 Incorporation of prior information 189 6.2 Examples on the use of the correlation coefficient 189 6.2.1 Use of the hyperbolic tangent transformation 189 6.2.2 Combination of several correlation coefficients 189 6.2.3 The squared correlation coefficient 190 6.3 Regression and the bivariate normal model 190 6.3.1 The model 190 6.3.2 Bivariate linear regression 191 6.3.3 Example 193 6.3.4 Case of known variance 194 6.3.5 The mean value at a given value of the explanatory variable 194 6.3.6 Prediction of observations at a given value of the explanatory variable 195 6.3.7 Continuation of the example 195 6.3.8 Multiple regression 196 6.3.9 Polynomial regression 196 6.4 Conjugate prior for the bivariate regression model 197 6.4.1 The problem of updating a regression line 197 6.4.2 Formulae for recursive construction of a regression line 197 6.4.3 Finding an appropriate prior 199 6.5 Comparison of several means – the one way model 200 6.5.1 Description of the one way layout 200 6.5.2 Integration over the nuisance parameters 201 6.5.3 Derivation of the F distribution 203 6.5.4 Relationship to the analysis of variance 203 6.5.5 Example 204 6.5.6 Relationship to a simple linear regression model 206 6.5.7 Investigation of contrasts 207 6.6 The two way layout 209 6.6.1 Notation 209 6.6.2 Marginal posterior distributions 210 6.6.3 Analysis of variance 212 6.7 The general linear model 212 6.7.1 Formulation of the general linear model 212 6.7.2 Derivation of the posterior 214 6.7.3 Inference for a subset of the parameters 215 6.7.4 Application to bivariate linear regression 216 6.8 Exercises on Chapter 6 217 7 Other topics 221 7.1 The likelihood principle 221 7.1.1 Introduction 221 7.1.2 The conditionality principle 222 7.1.3 The sufficiency principle 223 7.1.4 The likelihood principle 223 7.1.5 Discussion 225 7.2 The stopping rule principle 226 7.2.1 Definitions 226 7.2.2 Examples 226 7.2.3 The stopping rule principle 227 7.2.4 Discussion 228 7.3 Informative stopping rules 229 7.3.1 An example on capture and recapture of fish 229 7.3.2 Choice of prior and derivation of posterior 230 7.3.3 The maximum likelihood estimator 231 7.3.4 Numerical example 231 7.4 The likelihood principle and reference priors 232 7.4.1 The case of Bernoulli trials and its general implications 232 7.4.2 Conclusion 233 7.5 Bayesian decision theory 234 7.5.1 The elements of game theory 234 7.5.2 Point estimators resulting from quadratic loss 236 7.5.3 Particular cases of quadratic loss 237 7.5.4 Weighted quadratic loss 238 7.5.5 Absolute error loss 238 7.5.6 Zero-one loss 239 7.5.7 General discussion of point estimation 240 7.6 Bayes linear methods 240 7.6.1 Methodology 240 7.6.2 Some simple examples 241 7.6.3 Extensions 243 7.7 Decision theory and hypothesis testing 243 7.7.1 Relationship between decision theory and classical hypothesis testing 243 7.7.2 Composite hypotheses 245 7.8 Empirical Bayes methods 245 7.8.1 Von Mises’ example 245 7.8.2 The Poisson case 246 7.9 Exercises on Chapter 7 247 8 Hierarchical models 253 8.1 The idea of a hierarchical model 253 8.1.1 Definition 253 8.1.2 Examples 254 8.1.3 Objectives of a hierarchical analysis 257 8.1.4 More on empirical Bayes methods 257 8.2 The hierarchical normal model 258 8.2.1 The model 258 8.2.2 The Bayesian analysis for known overall mean 259 8.2.3 The empirical Bayes approach 261 8.3 The baseball example 262 8.4 The Stein estimator 264 8.4.1 Evaluation of the risk of the James–Stein estimator 267 8.5 Bayesian analysis for an unknown overall mean 268 8.5.1 Derivation of the posterior 270 8.6 The general linear model revisited 272 8.6.1 An informative prior for the general linear model 272 8.6.2 Ridge regression 274 8.6.3 A further stage to the general linear model 275 8.6.4 The one way model 276 8.6.5 Posterior variances of the estimators 277 8.7 Exercises on Chapter 8 277 9 The Gibbs sampler and other numerical methods 281 9.1 Introduction to numerical methods 281 9.1.1 Monte Carlo methods 281 9.1.2 Markov chains 282 9.2 The EM algorithm 283 9.2.1 The idea of the EM algorithm 283 9.2.2 Why the EM algorithm works 285 9.2.3 Semi-conjugate prior with a normal likelihood 287 9.2.4 The EM algorithm for the hierarchical normal model 288 9.2.5 A particular case of the hierarchical normal model 290 9.3 Data augmentation by Monte Carlo 291 9.3.1 The genetic linkage example revisited 291 9.3.2 Use of R 291 9.3.3 The genetic linkage example in R 292 9.3.4 Other possible uses for data augmentation 293 9.4 The Gibbs sampler 294 9.4.1 Chained data augmentation 294 9.4.2 An example with observed data 296 9.4.3 More on the semi-conjugate prior with a normal likelihood 299 9.4.4 The Gibbs sampler as an extension of chained data augmentation 301 9.4.5 An application to change-point analysis 302 9.4.6 Other uses of the Gibbs sampler 306 9.4.7 More about convergence 309 9.5 Rejection sampling 311 9.5.1 Description 311 9.5.2 Example 311 9.5.3 Rejection sampling for log-concave distributions 311 9.5.4 A practical example 313 9.6 The Metropolis–Hastings algorithm 317 9.6.1 Finding an invariant distribution 317 9.6.2 The Metropolis–Hastings algorithm 318 9.6.3 Choice of a candidate density 320 9.6.4 Example 321 9.6.5 More realistic examples 322 9.6.6 Gibbs as a special case of Metropolis–Hastings 322 9.6.7 Metropolis within Gibbs 323 9.7 Introduction to WinBUGS and OpenBUGS 323 9.7.1 Information about WinBUGS and OpenBUGS 323 9.7.2 Distributions in WinBUGS and OpenBUGS 324 9.7.3 A simple example using WinBUGS 324 9.7.4 The pump failure example revisited 327 9.7.5 DoodleBUGS 327 9.7.6 coda 329 9.7.7 R2WinBUGS and R2OpenBUGS 329 9.8 Generalized linear models 332 9.8.1 Logistic regression 332 9.8.2 A general framework 334 9.9 Exercises on Chapter 9 335 10 Some approximate methods 340 10.1 Bayesian importance sampling 340 10.1.1 Importance sampling to find HDRs 343 10.1.2 Sampling importance re-sampling 344 10.1.3 Multidimensional applications 344 10.2 Variational Bayesian methods: simple case 345 10.2.1 Independent parameters 347 10.2.2 Application to the normal distribution 349 10.2.3 Updating the mean 350 10.2.4 Updating the variance 351 10.2.5 Iteration 352 10.2.6 Numerical example 352 10.3 Variational Bayesian methods: general case 353 10.3.1 A mixture of multivariate normals 353 10.4 ABC: Approximate Bayesian Computation 356 10.4.1 The ABC rejection algorithm 356 10.4.2 The genetic linkage example 358 10.4.3 The ABC Markov Chain Monte Carlo algorithm 360 10.4.4 The ABC Sequential Monte Carlo algorithm 362 10.4.5 The ABC local linear regression algorithm 365 10.4.6 Other variants of ABC 366 10.5 Reversible jump Markov chain Monte Carlo 367 10.5.1 RJMCMC algorithm 367 10.6 Exercises on Chapter 10 369 Appendix A Common statistical distributions 373 A.1 Normal distribution 374 A.2 Chi-squared distribution 375 A.3 Normal approximation to chi-squared 376 A.4 Gamma distribution 376 A.5 Inverse chi-squared distribution 377 A.6 Inverse chi distribution 378 A.7 Log chi-squared distribution 379 A.8 Student’s t distribution 380 A.9 Normal/chi-squared distribution 381 A.10 Beta distribution 382 A.11 Binomial distribution 383 A.12 Poisson distribution 384 A.13 Negative binomial distribution 385 A.14 Hypergeometric distribution 386 A.15 Uniform distribution 387 A.16 Pareto distribution 388 A.17 Circular normal distribution 389 A.18 Behrens’ distribution 391 A.19 Snedecor’s F distribution 393 A.20 Fisher’s z distribution 393 A.21 Cauchy distribution 394 A.22 The probability that one beta variable is greater than another 395 A.23 Bivariate normal distribution 395 A.24 Multivariate normal distribution 396 A.25 Distribution of the correlation coefficient 397 Appendix B Tables 399 B.1 Percentage points of the Behrens–Fisher distribution 399 B.2 Highest density regions for the chi-squared distribution 402 B.3 HDRs for the inverse chi-squared distribution 404 B.4 Chi-squared corresponding to HDRs for log chi-squared 406 B.5 Values of F corresponding to HDRs for log F 408 Appendix C R programs 430 Appendix D Further reading 436 D.1 Robustness 436 D.2 Nonparametric methods 436 D.3 Multivariate estimation 436 D.4 Time series and forecasting 437 D.5 Sequential methods 437 D.6 Numerical methods 437 D.7 Bayesian networks 437 D.8 General reading 438 References 439 Index 455

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Book SynopsisFundamentals of Statistical Reasoning in Education 4th Edition, like the first three editions, is written largely with students of education in mind. Accordingly, Theodore Coladarci and Casey D. Cobb have drawn primarily on examples and issues found in school settings, such as those having to do with instruction, learning, motivation, and assessment. The emphasis on educational applications notwithstanding, the authors are confident that readers will find Fundamentals of Statistical Reasoning in Education 4th Edition of general relevance to other disciplines in the behavioral sciences as well. The 4th Edition of Fundamentals is still designed as a one semester book. The authors intentionally sidestep topics that few introductory courses cover (e.g., factorial analysis of variance, repeated measures analysis of variance, multiple regression). At the same time, effect size and confidence intervals are incorporated throughout, which today are regarded Trade Review"This book, like the first three editions, is written largely with students of education in mind. Accordingly, the authors have drawn primarily on examples and issues found in school settings, such as those having to do with instruction, learning, motivation, and assessment. The emphasis on educational applications notwithstanding, the authors are confident that readers will find this book of general relevance to other disciplines in the behavioral sciences as well." (Zentralblatt MATH 2016)Table of ContentsChapter 1 Introduction 1 1.1 Why Statistics? 1 1.2 Descriptive Statistics 2 1.3 Inferential Statistics 3 1.4 The Role of Statistics in Educational Research 4 1.5 Variables and Their Measurement 5 1.6 Some Tips on Studying Statistics 8 PART 1 DESCRIPTIVE STATISTICS 13 Chapter 2 Frequency Distributions 14 2.1 Why Organize Data? 14 2.2 Frequency Distributions for Quantitative Variables 14 2.3 Grouped Scores 15 2.4 Some Guidelines for Forming Class Intervals 17 2.5 Constructing a Grouped-Data Frequency Distribution 18 2.6 The Relative Frequency Distribution 19 2.7 Exact Limits 21 2.8 The Cumulative Percentage Frequency Distribution 22 2.9 Percentile Ranks 23 2.10 Frequency Distributions for Qualitative Variables 25 2.11 Summary 26 Chapter 3 Graphic Representation 34 3.1 Why Graph Data? 34 3.2 Graphing Qualitative Data: The Bar Chart 34 3.3 Graphing Quantitative Data: The Histogram 35 3.4 Relative Frequency and Proportional Area 39 3.5 Characteristics of Frequency Distributions 41 3.6 The Box Plot 44 3.7 Summary 45 Chapter 4 Central Tendency 52 4.1 The Concept of Central Tendency 52 4.2 The Mode 52 4.3 The Median 53 4.4 The Arithmetic Mean 54 4.5 Central Tendency and Distribution Symmetry 57 4.6 Which Measure of Central Tendency to Use? 59 4.7 Summary 59 Chapter 5 Variability 66 5.1 Central Tendency Is Not Enough: The Importance of Variability 66 5.2 The Range 67 5.3 Variability and Deviations From the Mean 68 5.4 The Variance 69 5.5 The Standard Deviation 70 5.6 The Predominance of the Variance and Standard Deviation 71 5.7 The Standard Deviation and the Normal Distribution 72 5.8 Comparing Means of Two Distributions: The Relevance of Variability 73 5.9 In the Denominator: n Versus n −1 75 5.10 Summary 76 Chapter 6 Normal Distributions and Standard Scores 81 6.1 A Little History: Sir Francis Galton and the Normal Curve 81 6.2 Properties of the Normal Curve 82 6.3 More on the Standard Deviation and the Normal Distribution 82 6.4 z Scores 84 6.5 The Normal Curve Table 87 6.6 Finding Area When the Score Is Known 88 6.7 Reversing the Process: Finding Scores When the Area Is Known 91 6.8 Comparing Scores From Different Distributions 93 6.9 Interpreting Effect Size 94 6.10 Percentile Ranks and the Normal Distribution 96 6.11 Other Standard Scores 97 6.12 Standard Scores Do Not “Normalize” a Distribution 98 6.13 The Normal Curve and Probability 98 6.14 Summary 99 Chapter 7 Correlation 106 7.1 The Concept of Association 106 7.2 Bivariate Distributions and Scatterplots 106 7.3 The Covariance 111 7.4 The Pearson r 117 7.5 Computation of r: The Calculating Formula 118 7.6 Correlation and Causation 120 7.7 Factors Influencing Pearson r 122 7.8 Judging the Strength of Association: r2 125 7.9 Other Correlation Coefficients 127 7.10 Summary 127 Chapter 8 Regression and Prediction 134 8.1 Correlation Versus Prediction 134 8.2 Determining the Line of Best Fit 135 8.3 The Regression Equation in Terms of Raw Scores 138 8.4 Interpreting the Raw-Score Slope 141 8.5 The Regression Equation in Terms of z Scores 141 8.6 Some Insights Regarding Correlation and Prediction 142 8.7 Regression and Sums of Squares 145 8.8 Residuals and Unexplained Variation 147 8.9 Measuring the Margin of Prediction Error: The Standard Error of Estimate 148 8.10 Correlation and Causality (Revisited) 152 8.11 Summary 153 PART 2 INFERENTIAL STATISTICS 163 Chapter 9 Probability and Probability Distributions 164 9.1 Statistical Inference: Accounting for Chance in Sample Results 164 9.2 Probability: The Study of Chance 165 9.3 Definition of Probability 166 9.4 Probability Distributions 168 9.5 The OR/addition Rule 169 9.6 The AND/Multiplication Rule 171 9.7 The Normal Curve as a Probability Distribution 172 9.8 “So What?”—Probability Distributions as the Basis for Statistical Inference 174 9.9 Summary 175 Chapter 10 Sampling Distributions 179 10.1 From Coins to Means 179 10.2 Samples and Populations 180 10.3 Statistics and Parameters 181 10.4 Random Sampling Model 181 10.5 Random Sampling in Practice 183 10.6 Sampling Distributions of Means 184 10.7 Characteristics of a Sampling Distribution of Means 185 10.8 Using a Sampling Distribution of Means to Determine Probabilities 188 10.9 The Importance of Sample Size (n) 191 10.10 Generality of the Concept of a Sampling Distribution 193 10.11 Summary 193 Chapter 11 Testing Statistical Hypotheses About μ When σ Is Known: The One-Sample z Test 199 11.1 Testing a Hypothesis About μ: Does “Homeschooling” Make a Difference? 199 11.2 Dr. Meyer’s Problem in a Nutshell 200 11.3 The Statistical Hypotheses: H0 and H1 201 11.4 The Test Statistic z 202 11.5 The Probability of the Test Statistic: The p Value 203 11.6 The Decision Criterion: Level of Significance (α) 204 11.7 The Level of Significance and Decision Error 207 11.8 The Nature and Role of H0 and H1 209 11.9 Rejection Versus Retention of H0 209 11.10 Statistical Significance Versus Importance 210 11.11 Directional and Nondirectional Alternative Hypotheses 212 11.12 The Substantive Versus the Statistical 214 11.13 Summary 215 Chapter 12 Estimation 222 12.1 Hypothesis Testing Versus Estimation 222 12.2 Point Estimation Versus Interval Estimation 223 12.3 Constructing an Interval Estimate of μ 224 12.4 Interval Width and Level of Confidence 226 12.5 Interval Width and Sample Size 227 12.6 Interval Estimation and Hypothesis Testing 228 12.7 Advantages of Interval Estimation 230 12.8 Summary 230 Chapter 13 Testing Statistical Hypotheses About μ When σ Is Not Known: The One-Sample t Test 235 13.1 Reality: σ Often Is Unknown 235 13.2 Estimating the Standard Error of the Mean 236 13.3 The Test Statistic t 237 13.4 Degrees of Freedom 238 13.5 The Sampling Distribution of Student’s t 239 13.6 An Application of Student’s t 242 13.7 Assumption of Population Normality 244 13.8 Levels of Significance Versus p Values 244 13.9 Constructing a Confidence Interval for μ When σ Is Not Known 246 13.10 Summary 247 Chapter 14 Comparing the Means of Two Populations: Independent Samples 253 14.1 From One Mu (μ) to Two 253 14.2 Statistical Hypotheses 254 14.3 The Sampling Distribution of Differences Between Means 255 14.4 Estimating σx̄1-x̄2 257 14.5 The t Test for Two Independent Samples 259 14.6 Testing Hypotheses About Two Independent Means: An Example 260 14.7 Interval Estimation of μ1 − μ2 262 14.8 Appraising the Magnitude of a Difference: Measures of Effect Size for − 264 14.9 How Were Groups Formed? The Role of Randomization 268 14.10 Statistical Inferences and Nonstatistical Generalizations 269 14.11 Summary 270 Chapter 15 Comparing the Means of Dependent Samples 278 15.1 The Meaning of “Dependent” 278 15.2 Standard Error of the Difference Between Dependent Means 279 15.3 Degrees of Freedom 281 15.4 The t Test for Two Dependent Samples 281 15.5 Testing Hypotheses About Two Dependent Means: An Example 283 15.6 Interval Estimation of μD 286 15.7 Summary 287 Chapter 16 Comparing the Means of Three or More Independent Samples: One-Way Analysis of Variance 294 16.1 Comparing More Than Two Groups: Why Not Multiplet Tests? 294 16.2 The Statistical Hypotheses in One-Way ANOVA 295 16.3 The Logic of One-Way ANOVA: An Overview 296 16.4 Alison’s Reply to Gregory 299 16.5 Partitioning the Sums of Squares 300 16.6 Within-Groups and Between- Groups Variance Estimates 303 16.7 The F Test 304 16.8 Tukey’s “HSD” Test 306 16.9 Interval Estimation of μi − μj 308 16.10 One-Way ANOVA: Summarizing the Steps 309 16.11 Estimating the Strength of the Treatment Effect: Effect Size (ω2) 311 16.12 ANOVA Assumptions (and Other Considerations) 312 16.13 Summary 313 Chapter 17 Inferences about the Pearson Correlation Coefficient 322 17.1 From μ to ρ 322 17.2 The Sampling Distribution of r When ρ = 0 322 17.3 Testing the Statistical Hypothesis That ρ = 0 324 17.4 An Example 324 17.5 In Brief: Student’s t Distribution and the Regression Slope (b) 326 17.6 Table E 326 17.7 The Role of n in the Statistical Significance of r 328 17.8 Statistical Significance Versus Importance (Again) 329 17.9 Testing Hypotheses Other Than ρ = 0 329 17.10 Interval Estimation of ρ 330 17.11 Summary 332 Chapter 18 Making Inferences From Frequency Data 338 18.1 Frequency Data Versus Score Data 338 18.2 A Problem Involving Frequencies: The One-Variable Case 339 18.3 χ2: A Measure of Discrepancy Between Expected and Observed Frequencies 340 18.4 The Sampling Distribution of χ2 341 18.5 Completion of the Voter Survey Problem: The χ2 Goodness-of-Fit Test 343 18.6 The χ2 Test of a Single Proportion 344 18.7 Interval Estimate of a Single Proportion 345 18.8 When There Are Two Variables: The χ2 Test of Independence 347 18.9 Finding Expected Frequencies in the Two-Variable Case 348 18.10 Calculating the Two-Variable χ2 350 18.11 The χ2 Test of Independence: Summarizing the Steps 351 18.12 The 2 × 2 Contingency Table 352 18.13 Testing a Difference Between Two Proportions 353 18.14 The Independence of Observations 353 18.15 χ2 and Quantitative Variables 354 18.16 Other Considerations 355 18.17 Summary 355 Chapter 19 Statistical “Power” (and How to Increase It) 363 19.1 The Power of a Statistical Test 363 19.2 Power and Type II Error 364 19.3 Effect Size (Revisited) 365 19.4 Factors Affecting Power: The Effect Size 366 19.5 Factors Affecting Power: Sample Size 367 19.6 Additional Factors Affecting Power 368 19.7 Significance Versus Importance 369 19.8 Selecting an Appropriate Sample Size 370 19.9 Summary 373 Epilogue A Note on (Almost) Assumption-Free Tests 379 References 380 Appendix A Review of Basic Mathematics 382 A.1 Introduction 382 A.2 Symbols and Their Meaning 382 A.3 Arithmetic Operations Involving Positive and Negative Numbers 383 A.4 Squares and Square Roots 383 A.5 Fractions 384 A.6 Operations Involving Parentheses 385 A.7 Approximate Numbers, Computational Accuracy, and Rounding 386 Appendix B Answers to Selected End-of-Chapter Problems 387 Appendix C Statistical Tables 408 Glossary 421 Index 427 Useful Formulas 433

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Book SynopsisThe 5th edition of Model Building in Mathematical Programming discusses the general principles of model building in mathematical programming and demonstrates how they can be applied by using several simplified but practical problems from widely different contexts. Suggested formulations and solutions are given together with some computational experience to give the reader a feel for the computational difficulty of solving that particular type of model. Furthermore, this book illustrates the scope and limitations of mathematical programming, and shows how it can be applied to real situations. By emphasizing the importance of the building and interpreting of models rather than the solution process, the author attempts to fill a gap left by the many works which concentrate on the algorithmic side of the subject. In this article, H.P. Williams explains his original motivation and objectives in writing the book, how it has been modified and updated over the years, whTable of ContentsPreface PART 1 1 Introduction 1.1 The Concept of a Model 1.2 Mathematical Programming Models 2 Solving Mathematical Programming Models 2.1 Algorithms and Packages 2.2 Practical Considerations 2.3 Decision Support and Expert Systems 2.4 Constraint Programming 3 Building Linear Programming Models 3.1 The Importance of Linearity 3.2 Defining Objectives 3.3 Defining Constraints 3.4 How to Build a Good Model 3.5 The Use of Modelling Languages 4 Structured Linear Programming Models 4.1 Multiple Plant, Product, and Period Models 4.2 Stochastic Programming Models 4.3 Decomposing a Large Model 5 Applications and Special Types of Mathematical Programming Model 5.1 Typical Applications 5.2 Economic Models 5.3 Network Models 5.4 Converting Linear Programs to Networks 6 Interpreting and Using the Solution of a Linear Programming Model 6.1 Validating a Model 6.2 Economic Interpretations 6.3 Sensitivity Analysis and the Stability of a Model 6.4 Further Investigations Using a Model 6.5 Presentation of the Solutions 7 Non-linear Models 7.1 Typical Applications 7.2 Local and Global Optima 7.3 Separable Programming 7.4 Converting a Problem to a Separable Model 8 Integer Programming 8.1 Introduction 8.2 The Applicability of Integer Programming 8.3 Solving Integer Programming Models 9 Building Integer Programming Models I 9.1 The Uses of Discrete Variables 9.2 Logical Conditions and Zero—One Variables 9.3 Special Ordered Sets of Variables 9.4 Extra Conditions Applied to Linear Programming Models 9.5 Special Kinds of Integer Programming Model 9.6 Column Generation 10 Building Integer Programming Models II 10.1 Good and Bad Formulations 10.2 Simplifying an Integer Programming Model 10.3 Economic Information Obtainable by Integer Programming 10.4 Sensitivity Analysis and the Stability of a Model 10.5 When and How to Use Integer Programming 11 The Implementation of a Mathematical Programming System of Planning 11.1 Acceptance and Implementation 11.2 The Unification of Organizational Functions 11.3 Centralization versus Decentralization 11.4 The Collection of Data and the Maintenance of a Model PART 2 12 The Problems 12.1 Food Manufacture 1 When to buy and how to blend 12.2 Food Manufacture 2 Limiting the number of ingredients and adding extra conditions 12.3 Factory Planning 1 What to make, on what machines, and when 12.4 Factory Planning 2 When should machines be down for maintenance 12.5 Manpower Planning How to recruit, retrain, make redundant, or overman 12.6 Refinery Optimization How to run an oil refinery 12.7 Mining Which pits to work and when to close them down 12.8 Farm Planning How much to grow and rear 12.9 Economic Planning How should an economy grow 12.10 Decentralization How to disperse offices from the capital 12.11 Curve Fitting Fitting a curve to a set of data points 12.12 Logical Design Constructing an electronic system with a minimum number of components 12.13 Market Sharing Assigning retailers to company divisions 12.14 Opencast Mining How much to excavate 12.15 Tariff Rates (Power Generation) How to determine tariff rates for the sale of electricity 12.16 Hydro Power How to generate and combine hydro and thermal electricity generation 12.17 Three-dimensional Noughts and Crosses A combinatorial problem 12.18 Optimizing a Constraint Reconstructing an integer programming constraint more simply 12.19 Distribution 1 Which factories and depots to supply which customers 12.20 Depot Location (Distribution 2) Where should new depots be built 12.21 Agricultural Pricing What prices to charge for dairy products 12.22 Efficiency Analysis How to use data envelopment analysis to compare efficiencies of garages 12.23 Milk Collection How to route and assign milk collection lorries to farms 12.24 Yield Management What quantities of airline tickets to sell at what prices and what times 12.25 Car Rental 1 How many cars to own and where to locate them 12.26 Car Rental 2 Where should repair capacity be increased 12.27 Lost Baggage Distribution Which vehicles should go to which customers and in what order 12.28 Protein Folding How a string of Amino Acids is likely to fold 12.29 Protein Comparison How similar are two proteins PART 3 13 Formulation and Discussion of Problems 13.1 Food Manufacture 1 13.2 Food Manufacture 2 13.3 Factory Planning 1 13.4 Factory Planning 2 13.5 Manpower Planning 13.6 Refinery Optimization 13.7 Mining 13.8 Farm Planning 13.9 Economic Planning 13.10 Decentralization 13.11 Curve Fitting 13.12 Logical Design 13.13 Market Sharing 13.14 Opencast Mining 13.15 Tariff Rates (Power Generation) 13.16 Hydro Power 13.17 Three-dimensional Noughts and Crosses 13.18 Optimizing a Constraint 13.19 Distribution 1 13.20 Depot Location (Distribution 2) 13.21 Agricultural Pricing 13.22 Efficiency Analysis 13.23 Milk Collection 13.24 Yield Management 13.25 Car Rental 1 13.26 Car Rental 2 13.27 Lost Baggage Distribution 13.28 Protein Folding 13.29 Protein Comparison PART 4 14 Solutions to Problems 14.1 Food Manufacture 1 14.2 Food Manufacture 2 14.3 Factory Planning 1 14.4 Factory Planning 2 14.5 Manpower Planning 14.6 Refinery Optimization 14.7 Mining 14.8 Farm Planning 14.9 Economic Planning 14.10 Decentralization 14.11 Curve Fitting 14.12 Logical Design 14.13 Market Sharing 14.14 Opencast Mining 14.15 Tariff Rates (Power Generation) 14.16 Hydro Power 14.17 Three-dimensional Noughts and Crosses 14.18 Optimizing a Constraint 14.19 Distribution 1 14.20 Depot Location (Distribution 2) 14.21 Agricultural Pricing 14.22 Efficiency Analysis 14.23 Milk Collection 14.24 Yield Management 14.25 Car Rental 1 14.26 Car Rental 2 14.27 Lost Baggage Distribution 14.28 Protein Folding 14.29 Protein Comparison References Author Index Subject Index

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Book SynopsisThe easy way to brush up on the math skills you need in real life Not everyone retains the math they learned in school. Like any skill, your ability to speak "math" can deteriorate if left unused.Table of ContentsIntroduction 1 Part I: Boning Up on Math Basics 7 Chapter 1: Awesome Operations: Math Fundamentals 9 Chapter 2: High School Reunion: Revisiting Key Principles of Algebra and Geometry 27 Chapter 3: Becoming a Believer: Conversion, Statistics, Probability and More 47 Chapter 4: The Miracle of Mental Math 59 Part II: Math for Everyday Activities 71 Chapter 5: Let’s Make a Deal! Math You Use When Shopping 73 Chapter 6: Mmm, Mmm, Good: Kitchen Calculations 97 Chapter 7: It Does a Body Good: Math for Health and Well-Being 109 Chapter 8: Putting Geometry to Work at Home 129 Chapter 9: Math and Statistics around Town and on the Road 141 Part III: Math to Manage Your Personal Finances 161 Chapter 10: Budgets, Bank Accounts, Credit Cards, and More 163 Chapter 11: Key Principles of Investment Math 183 Chapter 12: Covering Your Assets: Insurance Math 203 Chapter 13: Taking Math to Work 215 Chapter 14: How Taxing! (Almost) Understanding the Government 229 Part IV: The Part of Tens 241 Chapter 15: Ten Quick Calculations You Can Do in Your Head 243 Chapter 16: Ten Activities That Build Math Skills 249 Index 255

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Book SynopsisMultivariate Analysis Comprehensive Reference Work on Multivariate Analysis and its Applications The first edition of this book, by Mardia, Kent and Bibby, has been used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments. A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of: Basic properties of random vectors, copulas, normal distribution theory, and estimation Hypothesis testing, multivariate regression, and analysis of variance Principal component analysis, factor analysis, and canonical correlation analysis Discriminant analysis, cluster analysis, and multidimensional scaling New advances and techniques, including supervised and unsupervised statistical learning, graphical models and regularization methods for high-dimensional data Although primarily designed as a textbook for final year undergraduates and postgraduate students in mathematics and statistics, the book will also be of interest to research workers and applied scientists.

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Book SynopsisMathematics and Statistics for Financial Risk Management is a practical guide to modern financial risk management for both practitioners and academics. Now in its second edition with more topics, more sample problems and more real world examples, this popular guide to financial risk management introduces readers to practical quantitative techniques for analyzing and managing financial risk. In a concise and easy-to-read style, each chapter introduces a different topic in mathematics or statistics.As different techniques are introduced, sample problems and application sections demonstrate how these techniques can be applied to actual risk management problems. Exercises at the end of each chapter and the accompanying solutions at the end of the book allow readers to practice the techniques they are learning and monitor their progress.A companion Web site includes interactive Excel spreadsheet examples and templates. Mathematics and Statistics for FinanciTable of ContentsPreface ix What’s New in the Second Edition xi Acknowledgments xiii Chapter 1 Some Basic Math 1 Logarithms 1 Log Returns 2 Compounding 3 Limited Liability 4 Graphing Log Returns 5 Continuously Compounded Returns 6 Combinatorics 8 Discount Factors 9 Geometric Series 9 Problems 14 Chapter 2 Probabilities 15 Discrete Random Variables 15 Continuous Random Variables 15 Mutually Exclusive Events 21 Independent Events 22 Probability Matrices 22 Conditional Probability 24 Problems 26 Chapter 3 Basic Statistics 29 Averages 29 Expectations 34 Variance and Standard Deviation 39 Standardized Variables 41 Covariance 42 Correlation 43 Application: Portfolio Variance and Hedging 44 Moments 47 Skewness 48 Kurtosis 51 Coskewness and Cokurtosis 53 Best Linear Unbiased Estimator (BLUE) 57 Problems 58 Chapter 4 Distributions 61 Parametric Distributions 61 Uniform Distribution 61 Bernoulli Distribution 63 Binomial Distribution 65 Poisson Distribution 68 Normal Distribution 69 Lognormal Distribution 72 Central Limit Theorem 73 Application: Monte Carlo Simulations Part I: Creating Normal Random Variables 76 Chi-Squared Distribution 77 Student’s t Distribution 78 F-Distribution 79 Triangular Distribution 81 Beta Distribution 82 Mixture Distributions 83 Problems 86 Chapter 5 Multivariate Distributions and Copulas 89 Multivariate Distributions 89 Copulas 97 Problems 111 Chapter 6 Bayesian Analysis 113 Overview 113 Bayes’ Theorem 113 Bayes versus Frequentists 119 Many-State Problems 120 Continuous Distributions 124 Bayesian Networks 128 Bayesian Networks versus Correlation Matrices 130 Problems 132 Chapter 7 Hypothesis Testing and Confidence Intervals 135 Sample Mean Revisited 135 Sample Variance Revisited 137 Confidence Intervals 137 Hypothesis Testing 139 Chebyshev’s Inequality 142 Application: VaR 142 Problems 152 Chapter 8 Matrix Algebra 155 Matrix Notation 155 Matrix Operations 156 Application: Transition Matrices 163 Application: Monte Carlo Simulations Part II: Cholesky Decomposition 165 Problems 168 Chapter 9 Vector Spaces 169 Vectors Revisited 169 Orthogonality 172 Rotation 177 Principal Component Analysis 181 Application: The Dynamic Term Structure of Interest Rates 185 Application: The Structure of Global Equity Markets 191 Problems 193 Chapter 10 Linear Regression Analysis 195 Linear Regression (One Regressor) 195 Linear Regression (Multivariate) 203 Application: Factor Analysis 208 Application: Stress Testing 211 Problems 212 Chapter 11 Time Series Models 215 Random Walks 215 Drift-Diffusion Model 216 Autoregression 217 Variance and Autocorrelation 222 Stationarity 223 Moving Average 227 Continuous Models 228 Application: GARCH 230 Application: Jump-Diffusion Model 232 Application: Interest Rate Models 232 Problems 234 Chapter 12 Decay Factors 237 Mean 237 Variance 243 Weighted Least Squares 244 Other Possibilities 245 Application: Hybrid VaR 245 Problems 247 Appendix A Binary Numbers 249 Appendix B Taylor Expansions 251 Appendix C Vector Spaces 253 Appendix D Greek Alphabet 255 Appendix E Common Abbreviations 257 Appendix F Copulas 259 Answers 263 References 303 About the Author 305 About the Companion Website 307 Index 309

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Book SynopsisPraise for the Third Edition Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book. - MAA Reviews Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics. Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more aTrade Review"This is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics." (Springer Nature, 2016)Table of ContentsPREFACE xiii ACKNOWLEDGMENTS xv PART I METHODS 1 1 The Basic Method 3 1.1 The Probabilistic Method, 3 1.2 Graph Theory, 5 1.3 Combinatorics, 9 1.4 Combinatorial Number Theory, 11 1.5 Disjoint Pairs, 12 1.6 Independent Sets and List Coloring, 13 1.7 Exercises, 16 The Erd˝os–Ko–Rado Theorem, 18 2 Linearity of Expectation 19 2.1 Basics, 19 2.2 Splitting Graphs, 20 2.3 Two Quickies, 22 2.4 Balancing Vectors, 23 2.5 Unbalancing Lights, 25 2.6 Without Coin Flips, 26 2.7 Exercises, 27 Brégman’s Theorem, 29 3 Alterations 31 3.1 Ramsey Numbers, 31 3.2 Independent Sets, 33 3.3 Combinatorial Geometry, 34 3.4 Packing, 35 3.5 Greedy Coloring, 36 3.6 Continuous Time, 38 3.7 Exercises, 41 High Girth and High Chromatic Number, 43 4 The Second Moment 45 4.1 Basics, 45 4.2 Number Theory, 46 4.3 More Basics, 49 4.4 Random Graphs, 51 4.5 Clique Number, 55 4.6 Distinct Sums, 57 4.7 The Rödl nibble, 58 4.8 Exercises, 64 Hamiltonian Paths, 65 5 The Local Lemma 69 5.1 The Lemma, 69 5.2 Property B and Multicolored Sets of Real Numbers, 72 5.3 Lower Bounds for Ramsey Numbers, 73 5.4 A Geometric Result, 75 5.5 The Linear Arboricity of Graphs, 76 5.6 Latin Transversals, 80 5.7 Moser’s Fix-It Algorithm, 81 5.8 Exercises, 87 Directed Cycles, 88 6 Correlation Inequalities 89 6.1 The Four Functions Theorem of Ahlswede and Daykin, 90 6.2 The FKG Inequality, 93 6.3 Monotone Properties, 94 6.4 Linear Extensions of Partially Ordered Sets, 97 6.5 Exercises, 99 Turán’s Theorem, 100 7 Martingales and Tight Concentration 103 7.1 Definitions, 103 7.2 Large Deviations, 105 7.3 Chromatic Number, 107 7.4 Two General Settings, 109 7.5 Four Illustrations, 113 7.6 Talagrand’s Inequality, 116 7.7 Applications of Talagrand’s Inequality, 119 7.8 Kim–Vu Polynomial Concentration, 121 7.9 Exercises, 123 Weierstrass Approximation Theorem, 124 8 The Poisson Paradigm 127 8.1 The Janson Inequalities, 127 8.2 The Proofs, 129 8.3 Brun’s Sieve, 132 8.4 Large Deviations, 135 8.5 Counting Extensions, 137 8.6 Counting Representations, 139 8.7 Further Inequalities, 142 8.8 Exercises, 143 Local Coloring, 144 9 Quasirandomness 147 9.1 The Quadratic Residue Tournaments, 148 9.2 Eigenvalues and Expanders, 151 9.3 Quasirandom Graphs, 157 9.4 Szemerédi’s Regularity Lemma, 165 9.5 Graphons, 170 9.6 Exercises, 172 Random Walks, 174 PART II TOPICS 177 10 Random Graphs 179 10.1 Subgraphs, 180 10.2 Clique Number, 183 10.3 Chromatic Number, 184 10.4 Zero–One Laws, 186 10.5 Exercises, 193 Counting Subgraphs, 195 11 The Erd˝os–Rényi Phase Transition 197 11.1 An Overview, 197 11.2 Three Processes, 199 11.3 The Galton–Watson Branching Process, 201 11.4 Analysis of the Poisson Branching Process, 202 11.5 The Graph Branching Model, 204 11.6 The Graph and Poisson Processes Compared, 205 11.7 The Parametrization Explained, 207 11.8 The Subcritical Regions, 208 11.9 The Supercritical Regimes, 209 11.10 The Critical Window, 212 11.11 Analogies to Classical Percolation Theory, 214 11.12 Exercises, 219 Long paths in the supercritical regime, 220 12 Circuit Complexity 223 12.1 Preliminaries, 223 12.2 Random Restrictions and Bounded-Depth Circuits, 225 12.3 More on Bounded-Depth Circuits, 229 12.4 Monotone Circuits, 232 12.5 Formulae, 235 12.6 Exercises, 236 Maximal Antichains, 237 13 Discrepancy 239 13.1 Basics, 239 13.2 Six Standard Deviations Suffice, 241 13.3 Linear and Hereditary Discrepancy, 245 13.4 Lower Bounds, 248 13.5 The Beck–Fiala Theorem, 250 13.6 Exercises, 251 Unbalancing Lights, 253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces, 256 14.2 Empty Triangles Determined by Points in the Plane, 257 14.3 Geometrical Realizations of Sign Matrices, 259 14.4 𝜖-Nets and VC-Dimensions of Range Spaces, 261 14.5 Dual Shatter Functions and Discrepancy, 266 14.6 Exercises, 269 Efficient Packing, 270 15 Codes, Games, and Entropy 273 15.1 Codes, 273 15.2 Liar Game, 276 15.3 Tenure Game, 278 15.4 Balancing Vector Game, 279 15.5 Nonadaptive Algorithms, 281 15.6 Half Liar Game, 282 15.7 Entropy, 284 15.8 Exercises, 289 An Extremal Graph, 291 16 Derandomization 293 16.1 The Method of Conditional Probabilities, 293 16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297 16.3 Exercises, 302 Crossing Numbers, Incidences, Sums and Products, 303 17 Graph Property Testing 307 17.1 Property Testing, 307 17.2 Testing Colorability, 308 17.3 Testing Triangle-Freeness, 312 17.4 Characterizing the Testable Graph Properties, 314 17.5 Exercises, 316 Turán Numbers and Dependent Random Choice, 317 Appendix A Bounding of Large Deviations 321 A.1 Chernoff Bounds, 321 A.2 Lower Bounds, 330 A.3 Exercises, 334 Triangle-Free Graphs Have Large Independence Numbers, 336 Appendix B Paul Erd˝os 339 B.1 Papers, 339 B.2 Conjectures, 341 B.3 On Erd˝os, 342 B.4 Uncle Paul, 343 The Rich Get Richer, 346 Appendix C Hints to Selected Exercises 349 REFERENCES 355 AUTHOR INDEX 367 SUBJECT INDEX 371

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Book SynopsisThe best-selling author of Big Data is back, this time with a unique and in-depth insight into how specific companies use big data. Big data is on the tip of everyone''s tongue. Everyone understands its power and importance, but many fail to grasp the actionable steps and resources required to utilise it effectively. This book fills the knowledge gap by showing how major companies are using big data every day, from an up-close, on-the-ground perspective. From technology, media and retail, to sport teams, government agencies and financial institutions, learn the actual strategies and processes being used to learn about customers, improve manufacturing, spur innovation, improve safety and so much more. Organised for easy dip-in navigation, each chapter follows the same structure to give you the information you need quickly. For each company profiled, learn what data was used, what problem it solved and the processes put it place to make it practical, as well as the techniTrade ReviewIt is refreshing to read a book whose author simply puts the big data hype into practice. Ultimately, it offers a comprehensive narrative of why and how data is transforming the way businesses operate. (Marginalia on Engagement, April 2016) "This book is a brilliant introduction to the concept of big data, perfect for anybody who would like to know what it's all about and how this can be of benefit." (Institute of Management Services Journal, June 2016) Warmly recommended (The Marketing Society, July 2016) Another excellent text from Marr (BCS, August 2016)Table of ContentsIntroduction 1 1 Walmart: How Big Data is Used to Drive Supermarket Performance 5 2 CERN: Unravelling the Secrets of the Universe With Big Data 11 3 Netflix: How Netflix Used Big Data to Give Us the Programmes We Want 17 4 Rolls-Royce: How Big Data is Used to Drive Success in Manufacturing 25 5 Shell: How Big Oil Uses Big Data 31 6 Apixio: How Big Data is Transforming Healthcare 37 7 Lotus F1 Team: How Big Data is Essential to the Success of Motorsport Teams 45 8 Pendleton & Son Butchers: Big Data For Small Business 51 9 US Olympic Women’s Cycling Team: How Big Data Analytics is Used to Optimize Athletes’ Performance 57 10 ZSL: Big Data in the Zoo and to Protect Animals 63 11 Facebook: How Facebook Use Big Data to Understand Customers 69 12 John Deere: How Big Data Can Be Applied On Farms 75 13 Royal Bank of Scotland: Using Big Data to Make Customer Service More Personal 81 14 LinkedIn: How Big Data is Used to Fuel Social Media Success 87 15 Microsoft: Bringing Big Data to the Masses 95 16 Acxiom: Fuelling Marketing With Big Data 103 17 US Immigration and Customs: How Big Data is Used to Keep Passengers Safe and Prevent Terrorism 111 18 Nest: Bringing the Internet of Things Into the Home 117 19 GE: How Big Data is Fuelling the Industrial Internet 125 20 Etsy: How Big Data is Used in a Crafty Way 131 21 Narrative Science: How Big Data is Used to Tell Stories 137 22 BBC: How Big Data is Used in the Media 143 23 Milton Keynes: How Big Data is Used to Create Smarter Cities 149 24 Palantir: How Big Data is Used to Help the CIA and to Detect Bombs in Afghanistan 157 25 Airbnb: How Big Data is Used to Disrupt the Hospitality Industry 163 26 Sprint: Profiling Audiences Using Mobile Network Data 169 27 Dickey’s Barbecue Pit: How Big Data is Used to Gain Performance Insights Into One of America’s Most Successful Restaurant Chains 175 28 Caesars: Big Data at the Casino 181 29 Fitbit: Big Data in the Personal Fitness Arena 189 30 Ralph Lauren: Big Data in the Fashion Industry 195 31 Zynga: Big Data in the Gaming Industry 199 32 Autodesk: How Big Data is Transforming the Software Industry 205 33 Walt Disney Parks and Resorts: How Big Data is Transforming Our Family Holidays 211 34 Experian: Using Big Data to Make Lending Decisions and to Crack Down On Identity Fraud 217 35 Transport for London: How Big Data is Used to Improve and Manage Public Transport in London 223 36 The US Government: Using Big Data to Run a Country 229 37 IBM Watson: Teaching Computers to Understand and Learn 237 38 Google: How Big Data is at the Heart of Google’s Business Model 243 39 Terra Seismic: Using Big Data to Predict Earthquakes 251 40 Apple: How Big Data is at the Centre of Their Business 255 41 Twitter: How Twitter and IBM Deliver Customer Insights From Big Data 261 42 Uber: How Big Data is at the Centre of Uber’s Transportation Business 267 43 Electronic Arts: Big Data in Video Gaming 273 44 Kaggle: Crowdsourcing Your Data Scientist 281 45 Amazon: How Predictive Analytics Are Used to Get a 360-Degree View of Consumers 287 Final Thoughts 293 About the Author 297 Acknowledgements 299 Index 301

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Book SynopsisThe grade-saving Algebra I companion, with hundreds of additional practice problems online Algebra I Workbook For Dummies is your solution to the Algebra brain-block. With hundreds of practice and example problems mapped to the typical high school Algebra class, you''ll crack the code in no time! Each problem includes a full explanation so you can see where you went wrongor rightevery step of the way. From fractions to FOIL and everything in between, this guide will help you grasp the fundamental concepts you''ll use in every other math class you''ll ever take. This new third edition includes access to an online test bank, where you''ll find bonus chapter quizzes to help you test your understanding and pinpoint areas in need of review. Whether you''re preparing for an exam or seeking a start-to-finish study aid, this workbook is your ticket to acing algebra. Master basic operations and properties to solve any problem Simplify expressions witTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 4 Part 1: Getting Down to the Nitty-Gritty on Basic Operations 5 Chapter 1: Deciphering Signs in Numbers 7 Assigning Numbers Their Place 7 Reading and Writing Absolute Value 9 Adding Signed Numbers 10 Making a Difference with Signed Numbers 11 Multiplying Signed Numbers 12 Dividing Signed Numbers 14 Answers to Problems on Signed Numbers 15 Chapter 2: Incorporating Algebraic Properties 17 Getting a Grip on Grouping Symbols 17 Distributing the Wealth 19 Making Associations Work 20 Computing by Commuting 21 Answers to Problems on Algebraic Properties 23 Chapter 3: Making Fractions and Decimals Behave 25 Converting Improper and Mixed Fractions 25 Finding Fraction Equivalences 27 Making Proportional Statements 28 Finding Common Denominators 30 Adding and Subtracting Fractions 31 Multiplying and Dividing Fractions 32 Simplifying Complex Fractions 35 Changing Fractions to Decimals and Vice Versa 36 Performing Operations with Decimals 38 Answers to Problems on Fractions 39 Chapter 4: Exploring Exponents 45 Multiplying and Dividing Exponentials 45 Raising Powers to Powers 47 Using Negative Exponents 49 Writing Numbers with Scientific Notation 50 Answers to Problems on Discovering Exponents 52 Chapter 5: Taming Rampaging Radicals 55 Simplifying Radical Expressions 55 Rationalizing Fractions 57 Arranging Radicals as Exponential Terms 58 Using Fractional Exponents 60 Simplifying Expressions with Exponents 61 Estimating Answers 63 Answers to Problems on Radicals 64 Chapter 6: Simplifying Algebraic Expressions 67 Adding and Subtracting Like Terms 68 Multiplying and Dividing Algebraically 69 Incorporating Order of Operations 70 Evaluating Expressions 71 Answers to Problems on Algebraic Expressions 74 Part 2: Changing the Format of Expressions 77 Chapter 7: Specializing in Multiplication Matters 79 Distributing One Factor over Many 79 Curses, FOILed again — or not 80 Squaring binomials 82 Multiplying the sum and difference of the same two terms 83 Cubing binomials 84 Creating the Sum and Difference of Cubes 85 Raising binomials to higher powers 86 Answers to Problems on Multiplying Expressions 88 Chapter 8: Dividing the Long Way to Simplify Algebraic Expressions 91 Dividing by a Monomial 91 Dividing by a Binomial 93 Dividing by Polynomials with More Terms 96 Simplifying Division Synthetically 97 Answers to Problems on Division 99 Chapter 9: Figuring on Factoring 103 Pouring Over Prime Factorizations 103 Factoring Out the Greatest Common Factor 105 Reducing Algebraic Fractions 106 Answers to Problems on Factoring Expressions 108 Chapter 10: Taking the Bite Out of Binomial Factoring 111 Factoring the Difference of Squares 112 Factoring Differences and Sums of Cubes 113 Making Factoring a Multiple Mission 114 Answers to Problems on Factoring 115 Chapter 11: Factoring Trinomials and Special Polynomials 117 Focusing First on the Greatest Common Factor (GCF) 118 “Un”wrapping the FOIL 119 Factoring Quadratic-Like Trinomials 121 Factoring Trinomials Using More than One Method 122 Factoring by Grouping .124 Putting All the Factoring Together 126 Answers to Problems on Factoring Trinomials and Other Expressions 128 Part 3: Seek and Ye Shall Find Solutions 131 Chapter 12: Lining Up Linear Equations 133 Using the Addition/Subtraction Property 133 Using the Multiplication/Division Property 135 Putting Several Operations Together 136 Solving Linear Equations with Grouping Symbols 138 Working It Out with Fractions 140 Solving Proportions 142 Answers to Problems on Solving Linear Equations 144 Chapter 13: Muscling Up to Quadratic Equations 151 Using the Square Root Rule 152 Solving by Factoring 153 Using the Quadratic Formula 155 Completing the Square 158 Dealing with Impossible Answers 159 Answers to Problems on Solving Quadratic Equations 161 Chapter 14: Yielding to Higher Powers 167 Determining How Many Possible Roots 168 Applying the Rational Root Theorem 169 Using the Factor/Root Theorem 170 Solving by Factoring 172 Solving Powers That Are Quadratic-Like 174 Answers to Problems on Solving Higher Power Equations 176 Chapter 15: Reeling in Radical and Absolute Value Equations 179 Squaring Both Sides to Solve Radical Equations 180 Doubling the Fun with Radical Equations .182 Solving Absolute Value Equations 183 Answers to Problems on Radical and Absolute Value Equations 185 Chapter 16: Getting Even with Inequalities 189 Using the Rules to Work on Inequality Statements 190 Rewriting Inequalities by Using Interval Notation 191 Solving Linear Inequalities 192 Solving Quadratic Inequalities 193 Dealing with Polynomial and Rational Inequalities 195 Solving Absolute Value Inequalities 196 Solving Complex Inequalities 198 Answers to Problems on Working with Inequalities 199 Part 4: Solving Story Problems and Sketching Graphs 203 Chapter 17: Facing Up to Formulas 205 Working with Formulas 206 Deciphering Perimeter, Area, and Volume 207 Using perimeter formulas to get around 207 Squaring off with area formulas 209 Working with volume formulas 211 Getting Interested in Using Percent 213 Answers to Problems on Using Formulas 215 Chapter 18: Making Formulas Work in Basic Story Problems 219 Applying the Pythagorean Theorem 220 Using Geometry to Solve Story Problems 221 Putting Distance, Rate, and Time in a Formula 224 Examining the distance-rate-time formula 224 Going the distance with story problems 226 Answers to Making Formulas Work in Basic Story Problems 228 Chapter 19: Relating Values in Story Problems 233 Tackling Age Problems 234 Tackling Consecutive Integer Problems 235 Working Together on Work Problems 238 Answers to Relating Values in Story Problems 240 Chapter 20: Measuring Up with Quality and Quantity Story Problems 243 Achieving the Right Blend with Mixtures Problems 244 Concocting the Correct Solution One Hundred Percent of the Time 246 Dealing with Money Problems 248 Answers to Problems on Measuring Up with Quality and Quantity 250 Chapter 21: Getting a Handle on Graphing 255 Thickening the Plot with Points 255 Sectioning Off by Quadrants 257 Using Points to Lay Out Lines 258 Graphing Lines with Intercepts 260 Computing Slopes of Lines 261 Graphing with the Slope-Intercept Form 263 Changing to the Slope-Intercept Form 265 Writing Equations of Lines 266 Picking on Parallel and Perpendicular Lines 267 Finding Distances between Points 268 Finding the Intersections of Lines 269 Graphing Parabolas and Circles 270 Graphing with Transformations 272 Answers to Problems on Graphing 275 Part 5: The Part of Tens 283 Chapter 22: Ten Common Errors That Get Noticed 285 Squaring a Negative or Negative of a Square 285 Squaring a Binomial 286 Operating on Radicals 286 Distributing a Negative Throughout 287 Fracturing Fractions 287 Raising a Power to a Power 288 Making Negative Exponents Flip 288 Making Sense of Reversing the Sense 288 Using the Slope Formula Correctly 289 Writing Several Fractions as One 289 Chapter 23: Ten Quick Tips to Make Algebra a Breeze 291 Flipping Proportions 291 Multiplying Through to Get Rid of Fractions 292 Zeroing In on Fractions 292 Finding a Common Denominator 292 Dividing by 3 or 9 293 Dividing by 2, 4, or 8 293 Commuting Back and Forth 293 Factoring Quadratics 294 Making Radicals Less Rad, Baby 294 Applying Acronyms 294 Index 295

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Book SynopsisThe easy way to conquer calculus Calculus is hardno doubt about itand students often need help understanding or retaining the key concepts covered in class. Calculus Workbook For Dummies serves up the concept review and practice problems with an easy-to-follow, practical approach. Plus, you'll get free access to a quiz for every chapter online. With a wide variety of problems on everything covered in calculus class, you'll find multiple examples of limits, vectors, continuity, differentiation, integration, curve-sketching, conic sections, natural logarithms, and infinite series.Plus, you'll get hundreds of practice opportunities with detailed solutions that will help you master the math that is critical for scoring your highest in calculus. Review key conceptsTake hundreds of practice problemsGet access to free chapter quizzes onlineUse as a classroom supplement or with a tutor Get ready to quickly and easily increase your confidence and improve your skills in calculus.Table of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Pre-Calculus Review 5 Chapter 1: Getting Down to Basics: Algebra and Geometry 7 Fraction Frustration 7 Misc. Algebra: You Know, Like Miss South Carolina 9 Geometry: When Am I Ever Going to Need It? 11 Solutions for This Easy, Elementary Stuff 16 Chapter 2: Funky Functions and Tricky Trig 25 Figuring Out Your Functions 25 Trigonometric Calisthenics 29 Solutions to Functions and Trigonometry 33 Part 2: Limits and Continuity 41 Chapter 3: A Graph Is Worth a Thousand Words: Limits and Continuity 43 Digesting the Definitions: Limit and Continuity 44 Taking a Closer Look: Limit and Continuity Graphs 46 Solutions for Limits and Continuity 50 Chapter 4: Nitty-Gritty Limit Problems 53 Solving Limits with Algebra 54 Pulling Out Your Calculator: Useful “Cheating” 59 Making Yourself a Limit Sandwich 61 Into the Great Beyond: Limits at Infinity 63 Solutions for Problems with Limits 67 Part 3: Differentiation 77 Chapter 5: Getting the Big Picture: Differentiation Basics 79 The Derivative: A Fancy Calculus Word for Slope and Rate 79 The Handy-Dandy Difference Quotient 81 Solutions for Differentiation Basics 84 Chapter 6: Rules, Rules, Rules: The Differentiation Handbook 89 Rules for Beginners 89 Giving It Up for the Product and Quotient Rules 92 Linking Up with the Chain Rule 94 What to Do with Y’s: Implicit Differentiation 98 Getting High on Calculus: Higher Order Derivatives 101 Solutions for Differentiation Problems 103 Chapter 7: Analyzing Those Shapely Curves with the Derivative 117 The First Derivative Test and Local Extrema 117 The Second Derivative Test and Local Extrema 120 Finding Mount Everest: Absolute Extrema 122 Smiles and Frowns: Concavity and Inflection Points 126 The Mean Value Theorem: Go Ahead, Make My Day 129 Solutions for Derivatives and Shapes of Curves 131 Chapter 8: Using Differentiation to Solve Practical Problems 147 Optimization Problems: From Soup to Nuts 147 Problematic Relationships: Related Rates 150 A Day at the Races: Position, Velocity, and Acceleration 153 Solutions to Differentiation Problem Solving 157 Chapter 9: Even More Practical Applications of Differentiation 173 Make Sure You Know Your Lines: Tangents and Normals 173 Looking Smart with Linear Approximation 177 Calculus in the Real World: Business and Economics 179 Solutions to Differentiation Problem Solving 183 Part 4: Integration and Infinite Series 191 Chapter 10: Getting into Integration 193 Adding Up the Area of Rectangles: Kid Stuff 193 Sigma Notation and Riemann Sums: Geek Stuff 196 Close Isn’t Good Enough: The Definite Integral and Exact Area 200 Finding Area with the Trapezoid Rule and Simpson’s Rule 202 Solutions to Getting into Integration 205 Chapter 11: Integration: Reverse Differentiation 213 The Absolutely Atrocious and Annoying Area Function 213 Sound the Trumpets: The Fundamental Theorem of Calculus 216 Finding Antiderivatives: The Guess-and-Check Method 219 The Substitution Method: Pulling the Switcheroo 221 Solutions to Reverse Differentiation Problems 225 Chapter 12: Integration Rules for Calculus Connoisseurs 229 Integration by Parts: Here’s How u du It 229 Transfiguring Trigonometric Integrals 233 Trigonometric Substitution: It’s Your Lucky Day! 235 Partaking of Partial Fractions 237 Solutions for Integration Rules 241 Chapter 13: Who Needs Freud? Using the Integral to Solve Your Problems 255 Finding a Function’s Average Value 255 Finding the Area between Curves 256 Volumes of Weird Solids: No, You’re Never Going to Need This 258 Arc Length and Surfaces of Revolution 265 Solutions to Integration Application Problems 268 Chapter 14: Infinite (Sort of) Integrals 277 Getting Your Hopes Up with L’Hôpital’s Rule 278 Disciplining Those Improper Integrals 280 Solutions to Infinite (Sort of) Integrals 283 Chapter 15: Infinite Series: Welcome to the Outer Limits 287 The Nifty nth Term Test 287 Testing Three Basic Series 289 Apples and Oranges . . . and Guavas: Three Comparison Tests 291 Ratiocinating the Two “R” Tests 295 He Loves Me, He Loves Me Not: Alternating Series 297 Solutions to Infinite Series 299 Part 5: The Part of Tens 309 Chapter 16: Ten Things about Limits, Continuity, and Infinite Series 311 The 33333 Mnemonic 311 First 3 over the “l”: 3 parts to the definition of a limit 312 Fifth 3 over the “l”: 3 cases where a limit fails to exist 312 Second 3 over the “i”: 3 parts to the definition of continuity 312 Fourth 3 over the “i”: 3 cases where continuity fails to exist 312 Third 3 over the “m”: 3 cases where a derivative fails to exist 313 The 13231 Mnemonic 313 First 1: The nth term test of divergence 313 Second 1: The nth term test of convergence for alternating series 313 First 3: The three tests with names 313 Second 3: The three comparison tests 314 The 2 in the middle: The two R tests 314 Chapter 17: Ten Things You Better Remember about Differentiation 315 The Difference Quotient 315 The First Derivative Is a Rate 315 The First Derivative Is a Slope 316 Extrema, Sign Changes, and the First Derivative 316 The Second Derivative and Concavity 316 Inflection Points and Sign Changes in the Second Derivative 316 The Product Rule 317 The Quotient Rule 317 Linear Approximation 317 “PSST,” Here’s a Good Way to Remember the Derivatives of Trig Functions 317 Index 319

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Book SynopsisHandbook and reference for industrial statisticians and system reliability engineers System Reliability Theory: Models, Statistical Methods, and Applications, Third Editionpresents an updated and revised look at system reliability theory, modeling, and analytical methods. The new edition is based on feedback to the second edition from numerous students, professors,researchers,and industries around the world. New sections and chapters are added together with new real-world industry examples,andstandards and problemsare revised and updated. System Reliability Theorycovers a broad and deep array of system reliability topics, including: In depth discussion of failures and failure modes The main system reliability assessment methods Common-cause failure modeling Deterioration modeling Maintenance modeling andassessmentusing Python code Bayesian probability and methods Life data analysis using RTable of ContentsPreface xxiii About the Companion Website xxix 1 Introduction 1 1.1 What is Reliability? 1 1.1.1 Service Reliability 2 1.1.2 Past and Future Reliability 3 1.2 The Importance of Reliability 3 1.2.1 Related Applications 4 1.3 Basic Reliability Concepts 6 1.3.1 Reliability 6 1.3.2 Maintainability and Maintenance 8 1.3.3 Availability 8 1.3.4 Quality 9 1.3.5 Dependability 9 1.3.6 Safety and Security 10 1.3.7 RAM and RAMS 10 1.4 Reliability Metrics 11 1.4.1 Reliability Metrics for a Technical Item 11 1.4.2 Reliability Metrics for a Service 12 1.5 Approaches to Reliability Analysis 12 1.5.1 The Physical Approach to Reliability 13 1.5.2 Systems Approach to Reliability 13 1.6 Reliability Engineering 15 1.6.1 Roles of the Reliability Engineer 16 1.6.2 Timing of Reliability Studies 17 1.7 Objectives, Scope, and Delimitations of the Book 17 1.8 Trends and Challenges 19 1.9 Standards and Guidelines 20 1.10 History of System Reliability 20 1.11 Problems 26 References 27 2 The Study Object and its Functions 31 2.1 Introduction 31 2.2 System and System Elements 31 2.2.1 Item 32 2.2.2 Embedded Item 33 2.3 Boundary Conditions 33 2.3.1 Closed and Open Systems 34 2.4 Operating Context 35 2.5 Functions and Performance Requirements 35 2.5.1 Functions 35 2.5.2 Performance Requirements 36 2.5.3 Classification of Functions 37 2.5.4 Functional Modeling and Analysis 38 2.5.5 Function Trees 38 2.5.6 SADT and IDEF 0 39 2.6 System Analysis 41 2.6.1 Synthesis 41 2.7 Simple, Complicated, and Complex Systems 42 2.8 System Structure Modeling 44 2.8.1 Reliability Block Diagram 44 2.8.2 Series Structure 46 2.8.3 Parallel Structure 46 2.8.4 Redundancy 47 2.8.5 Voted Structure 47 2.8.6 Standby Structure 48 2.8.7 More Complicated Structures 48 2.8.8 Two Different System Functions 49 2.8.9 Practical Construction of RBDs 50 2.9 Problems 51 References 52 3 Failures and Faults 55 3.1 Introduction 55 3.1.1 States and Transitions 56 3.1.2 Operational Modes 56 3.2 Failures 57 3.2.1 Failures in a State 58 3.2.2 Failures During Transition 59 3.3 Faults 60 3.4 Failure Modes 60 3.5 Failure Causes and Effects 62 3.5.1 Failure Causes 62 3.5.2 Proximate Causes and Root Causes 63 3.5.3 Hierarchy of Causes 64 3.6 Classification of Failures and Failure Modes 64 3.6.1 Classification According to Local Consequence 65 3.6.2 Classification According to Cause 65 3.6.3 Failure Mechanisms 70 3.6.4 Software Faults 71 3.6.5 Failure Effects 71 3.7 Failure/Fault Analysis 72 3.7.1 Cause and Effect Analysis 73 3.7.2 Root Cause Analysis 74 3.8 Problems 76 References 77 4 Qualitative System Reliability Analysis 79 4.1 Introduction 79 4.1.1 Deductive Versus Inductive Analysis 80 4.2 FMEA/FMECA 80 4.2.1 Types of FMECA 81 4.2.2 Objectives of FMECA 82 4.2.3 FMECA Procedure 83 4.2.4 Applications 87 4.3 Fault Tree Analysis 88 4.3.1 Fault Tree Symbols and Elements 88 4.3.2 Definition of the Problem and the Boundary Conditions 91 4.3.3 Constructing the Fault Tree 92 4.3.4 Identification of Minimal Cut and Path Sets 95 4.3.5 MOCUS 96 4.3.6 Qualitative Evaluation of the Fault Tree 98 4.3.7 Dynamic Fault Trees 101 4.4 Event Tree Analysis 103 4.4.1 Initiating Event 104 4.4.2 Safety Functions 105 4.4.3 Event Tree Construction 106 4.4.4 Description of Resulting Event Sequences 106 4.5 Fault Trees versus Reliability Block Diagrams 109 4.5.1 Recommendation 111 4.6 Structure Function 111 4.6.1 Series Structure 112 4.6.2 Parallel Structure 112 4.6.3 koon:G Structure 113 4.6.4 Truth Tables 114 4.7 System Structure Analysis 114 4.7.1 Single Points of Failure 115 4.7.2 Coherent Structures 115 4.7.3 General Properties of Coherent Structures 117 4.7.4 Structures Represented by Paths and Cuts 119 4.7.5 Pivotal Decomposition 123 4.7.6 Modules of Coherent Structures 124 4.8 Bayesian Networks 127 4.8.1 Illustrative Examples 128 4.9 Problems 131 References 138 5 Probability Distributions in Reliability Analysis 141 5.1 Introduction 141 5.1.1 State Variable 142 5.1.2 Time-to-Failure 142 5.2 A Dataset 143 5.2.1 Relative Frequency Distribution 143 5.2.2 Empirical Distribution and Survivor Function 144 5.3 General Characteristics of Time-to-Failure Distributions 145 5.3.1 Survivor Function 147 5.3.2 Failure Rate Function 148 5.3.3 Conditional Survivor Function 153 5.3.4 Mean Time-to-Failure 154 5.3.5 Additional Probability Metrics 155 5.3.6 Mean Residual Lifetime 157 5.3.7 Mixture of Time-to-Failure Distributions 160 5.4 Some Time-to-Failure Distributions 161 5.4.1 The Exponential Distribution 161 5.4.2 The Gamma Distribution 168 5.4.3 TheWeibull Distribution 173 5.4.4 The Normal Distribution 180 5.4.5 The Lognormal Distribution 183 5.4.6 Additional Time-to-Failure Distributions 188 5.5 Extreme Value Distributions 188 5.5.1 The Gumbel Distribution of the Smallest Extreme 190 5.5.2 The Gumbel Distribution of the Largest Extreme 191 5.5.3 TheWeibull Distribution of the Smallest Extreme 191 5.6 Time-to-Failure Models With Covariates 193 5.6.1 Accelerated Failure Time Models 194 5.6.2 The Arrhenius Model 195 5.6.3 Proportional Hazards Models 198 5.7 Additional Continuous Distributions 198 5.7.1 The Uniform Distribution 198 5.7.2 The Beta Distribution 199 5.8 Discrete Distributions 200 5.8.1 Binomial Situation 200 5.8.2 The Binomial Distribution 201 5.8.3 The Geometric Distribution 201 5.8.4 The Negative Binomial Distribution 202 5.8.5 The Homogeneous Poisson Process 203 5.9 Classes of Time-to-Failure Distributions 205 5.9.1 IFR and DFR Distributions 206 5.9.2 IFRA and DFRA Distributions 208 5.9.3 NBU and NWU Distributions 208 5.9.4 NBUE and NWUE Distributions 209 5.9.5 Some Implications 209 5.10 Summary of Time-to-Failure Distributions 210 5.11 Problems 210 References 218 6 System Reliability Analysis 221 6.1 Introduction 221 6.1.1 Assumptions 222 6.2 System Reliability 222 6.2.1 Reliability of Series Structures 223 6.2.2 Reliability of Parallel Structures 224 6.2.3 Reliability of koon Structures 225 6.2.4 Pivotal Decomposition 226 6.2.5 Critical Component 227 6.3 Nonrepairable Systems 228 6.3.1 Nonrepairable Series Structures 228 6.3.2 Nonrepairable Parallel Structures 230 6.3.3 Nonrepairable 2oo3 Structures 234 6.3.4 A Brief Comparison 235 6.3.5 Nonrepairable koon Structures 236 6.4 Standby Redundancy 237 6.4.1 Passive Redundancy, Perfect Switching, No Repairs 238 6.4.2 Cold Standby, Imperfect Switch, No Repairs 240 6.4.3 Partly Loaded Redundancy, Imperfect Switch, No Repairs 241 6.5 Single Repairable Items 242 6.5.1 Availability 243 6.5.2 Average Availability with Perfect Repair 244 6.5.3 Availability of a Single Item with Constant Failure and Repair Rates 246 6.5.4 Operational Availability 247 6.5.5 Production Availability 248 6.5.6 Punctuality 249 6.5.7 Failure Rate of Repairable Items 249 6.6 Availability of Repairable Systems 252 6.6.1 The MUT and MDT of Repairable Systems 253 6.6.2 Computation Based on Minimal Cut Sets 258 6.6.3 Uptimes and Downtimes for Reparable Systems 260 6.7 Quantitative Fault Tree Analysis 262 6.7.1 Terminology and Symbols 263 6.7.2 Delimitations and Assumptions 263 6.7.3 Fault Trees with a Single AND-Gate 264 6.7.4 Fault Tree with a Single OR-Gate 265 6.7.5 The Upper Bound Approximation Formula for Q0(t) 265 6.7.6 The Inclusion–Exclusion Principle 267 6.7.7 ROCOF of a Minimal Cut Parallel Structure 271 6.7.8 Frequency of the TOP Event 271 6.7.9 Binary Decision Diagrams 273 6.8 Event Tree Analysis 275 6.9 Bayesian Networks 277 6.9.1 Influence and Cause 278 6.9.2 Independence Assumptions 278 6.9.3 Conditional Probability Table 279 6.9.4 Conditional Independence 280 6.9.5 Inference and Learning 282 6.9.6 BN and Fault Tree Analysis 282 6.10 Monte Carlo Simulation 284 6.10.1 Random Number Generation 285 6.10.2 Monte Carlo Next Event Simulation 287 6.10.3 Simulation of Multicomponent Systems 289 6.11 Problems 291 References 296 7 Reliability Importance Metrics 299 7.1 Introduction 299 7.1.1 Objectives of Reliability Importance Metrics 300 7.1.2 Reliability Importance Metrics Considered 300 7.1.3 Assumptions and Notation 301 7.2 Critical Components 302 7.3 Birnbaum’s Metric for Structural Importance 304 7.4 Birnbaum’s Metric of Reliability Importance 305 7.4.1 Birnbaum’s Metric in Fault Tree Analysis 307 7.4.2 A Second Definition of Birnbaum’s Metric 308 7.4.3 A Third Definition of Birnbaum’s Metric 310 7.4.4 Computation of Birnbaum’s Metric for Structural Importance 312 7.4.5 Variants of Birnbaum’s Metric 312 7.5 Improvement Potential 313 7.5.1 Relation to Birnbaum’s Metric 314 7.5.2 A Variant of the Improvement Potential 314 7.6 Criticality Importance 315 7.7 Fussell–Vesely’s Metric 317 7.7.1 Derivation of Formulas for Fussell–Vesely’s Metric 317 7.7.2 Relationship to Other Metrics for Importance 320 7.8 Differential Importance Metric 323 7.8.1 Option 1 323 7.8.2 Option 2 324 7.9 Importance Metrics for Safety Features 326 7.9.1 Risk AchievementWorth 327 7.9.2 Risk ReductionWorth 329 7.9.3 Relationship with the Improvement Potential 330 7.10 Barlow–Proschan’s Metric 331 7.11 Problems 333 References 335 8 Dependent Failures 337 8.1 Introduction 337 8.1.1 Dependent Events and Variables 337 8.1.2 Correlated Variables 338 8.2 Types of Dependence 340 8.3 Cascading Failures 340 8.3.1 Tight Coupling 342 8.4 Common-Cause Failures 342 8.4.1 Multiple Failures that Are Not a CCF 344 8.4.2 Causes of CCF 344 8.4.3 Defenses Against CCF 345 8.5 CCF Models and Analysis 346 8.5.1 Explicit Modeling 347 8.5.2 Implicit Modeling 348 8.5.3 Modeling Approach 348 8.5.4 Model Assumptions 349 8.6 Basic Parameter Model 349 8.6.1 Probability of a Specific Multiplicity 350 8.6.2 Conditional Probability of a Specific Multiplicity 351 8.7 Beta-Factor Model 352 8.7.1 Relation to the BPM 354 8.7.2 Beta-Factor Model in System Analysis 354 8.7.3 Beta-Factor Model for Nonidentical Components 358 8.7.4 C-Factor Model 360 8.8 Multi-parameter Models 360 8.8.1 Binomial Failure Rate Model 360 8.8.2 Multiple Greek Letter Model 362 8.8.3 Alpha-Factor Model 364 8.8.4 Multiple Beta-Factor Model 365 8.9 Problems 366 References 368 9 Maintenance and Maintenance Strategies 371 9.1 Introduction 371 9.1.1 What is Maintenance? 372 9.2 Maintainability 372 9.3 Maintenance Categories 374 9.3.1 Completeness of a Repair Task 377 9.3.2 Condition Monitoring 377 9.4 Maintenance Downtime 378 9.4.1 Downtime Caused by Failures 379 9.4.2 Downtime of a Series Structure 381 9.4.3 Downtime of a Parallel Structure 381 9.4.4 Downtime of a General Structure 382 9.5 Reliability Centered Maintenance 382 9.5.1 What is RCM? 383 9.5.2 Main Steps of an RCM Analysis 384 9.6 Total Productive Maintenance 396 9.7 Problems 398 References 399 10 Counting Processes 401 10.1 Introduction 401 10.1.1 Counting Processes 401 10.1.2 Basic Concepts 406 10.1.3 Martingale Theory 408 10.1.4 Four Types of Counting Processes 409 10.2 Homogeneous Poisson Processes 410 10.2.1 Main Features of the HPP 411 10.2.2 Asymptotic Properties 412 10.2.3 Estimate and Confidence Interval 412 10.2.4 Sum and Decomposition of HPPs 413 10.2.5 Conditional Distribution of Failure Time 414 10.2.6 Compound HPPs 415 10.3 Renewal Processes 417 10.3.1 Basic Concepts 417 10.3.2 The Distribution of Sn 418 10.3.3 The Distribution of N(t) 420 10.3.4 The Renewal Function 421 10.3.5 The Renewal Density 423 10.3.6 Age and Remaining Lifetime 427 10.3.7 Bounds for the Renewal Function 431 10.3.8 Superimposed Renewal Processes 433 10.3.9 Renewal Reward Processes 434 10.3.10 Delayed Renewal Processes 436 10.3.11 Alternating Renewal Processes 438 10.4 Nonhomogeneous Poisson Processes 447 10.4.1 Introduction and Definitions 447 10.4.2 Some Results 449 10.4.3 Parametric NHPP Models 452 10.4.4 Statistical Tests of Trend 454 10.5 Imperfect Repair Processes 455 10.5.1 Brown and Proschan’s model 456 10.5.2 Failure Rate Reduction Models 458 10.5.3 Age Reduction Models 461 10.5.4 Trend Renewal Process 462 10.6 Model Selection 464 10.7 Problems 466 References 470 11 Markov Analysis 473 11.1 Introduction 473 11.1.1 Markov Property 475 11.2 Markov Processes 476 11.2.1 Procedure to Establish the Transition Rate Matrix 479 11.2.2 Chapman–Kolmogorov Equations 482 11.2.3 Kolmogorov Differential Equations 483 11.2.4 State Equations 484 11.3 Asymptotic Solution 487 11.3.1 System Performance Metrics 492 11.4 Parallel and Series Structures 495 11.4.1 Parallel Structures of Independent Components 495 11.4.2 Series Structures of Independent Components 497 11.4.3 Series Structure of Components Where Failure of One Component Prevents Failure of the Other 499 11.5 Mean Time to First System Failure 501 11.5.1 Absorbing States 501 11.5.2 Survivor Function 504 11.5.3 Mean Time to the First System Failure 505 11.6 Systems with Dependent Components 507 11.6.1 Common Cause Failures 508 11.6.2 Load-Sharing Systems 510 11.7 Standby Systems 512 11.7.1 Parallel System with Cold Standby and Perfect Switching 513 11.7.2 Parallel System with Cold Standby and Perfect Switching (Item A is the Main Operating Item) 515 11.7.3 Parallel System with Cold Standby and Imperfect Switching (Item A is the Main Operating Item) 517 11.7.4 Parallel System with Partly Loaded Standby and Perfect Switching (Item A is the Main Operating Item) 518 11.8 Markov Analysis in Fault Tree Analysis 519 11.8.1 Cut Set Information 520 11.8.2 System Information 521 11.9 Time-Dependent Solution 521 11.9.1 Laplace Transforms 522 11.10 Semi-Markov Processes 524 11.11 Multiphase Markov Processes 526 11.11.1 Changing the Transition Rates 526 11.11.2 Changing the Initial State 527 11.12 Piecewise Deterministic Markov Processes 528 11.12.1 Definition of PDMP 529 11.12.2 State Probabilities 529 11.12.3 A Specific Case 530 11.13 Simulation of a Markov Process 532 11.14 Problems 536 References 543 12 Preventive Maintenance 545 12.1 Introduction 545 12.2 Terminology and Cost Function 546 12.3 Time-Based Preventive Maintenance 548 12.3.1 Age Replacement 549 12.3.2 Block Replacement 553 12.3.3 P–F Intervals 557 12.4 Degradation Models 564 12.4.1 Remaining Useful Lifetime 565 12.4.2 Trend Models; Regression-Based Models 567 12.4.3 Models with Increments 569 12.4.4 Shock Models 571 12.4.5 Stochastic Processes with Discrete States 573 12.4.6 Failure Rate Models 574 12.5 Condition-Based Maintenance 574 12.5.1 CBM Strategy 575 12.5.2 Continuous Monitoring and Finite Discrete State Space 576 12.5.3 Continuous Monitoring and Continuous State Space 581 12.5.4 Inspection-Based Monitoring and Finite Discrete State Space 583 12.5.5 Inspection-Based Monitoring and Continuous State Space 586 12.6 Maintenance of Multi-Item Systems 587 12.6.1 System Model 587 12.6.2 Maintenance Models 589 12.6.3 An Illustrative Example 591 12.7 Problems 595 References 601 13 Reliability of Safety Systems 605 13.1 Introduction 605 13.2 Safety-Instrumented Systems 606 13.2.1 Main SIS Functions 607 13.2.2 Testing of SIS Functions 608 13.2.3 Failure Classification 609 13.3 Probability of Failure on Demand 611 13.3.1 Probability of Failure on Demand 612 13.3.2 Approximation Formulas 617 13.3.3 Mean Downtime in a Test Interval 618 13.3.4 Mean Number of Test Intervals Until First Failure 619 13.3.5 Staggered Testing 620 13.3.6 Nonnegligible Repair Time 621 13.4 Safety Unavailability 622 13.4.1 Probability of Critical Situation 623 13.4.2 Spurious Trips 623 13.4.3 Failures Detected by Diagnostic Self-Testing 625 13.5 Common Cause Failures 627 13.5.1 Diagnostic Self-Testing and CCFs 629 13.6 CCFs Between Groups and Subsystems 631 13.6.1 CCFs Between Voted Groups 632 13.6.2 CCFs Between Subsystems 632 13.7 IEC 61508 632 13.7.1 Safety Lifecycle 633 13.7.2 Safety Integrity Level 634 13.7.3 Compliance with IEC 61508 635 13.8 The PDS Method 638 13.9 Markov Approach 639 13.9.1 All Failures are Repaired After Each Test 643 13.9.2 All Critical Failures Are Repaired after Each Test 644 13.9.3 Imperfect Repair after Each Test 644 13.10 Problems 644 References 652 14 Reliability Data Analysis 655 14.1 Introduction 655 14.1.1 Purpose of the Chapter 656 14.2 Some Basic Concepts 656 14.2.1 Datasets 657 14.2.2 Survival Times 658 14.2.3 Categories of Censored Datasets 660 14.2.4 Field Data Collection Exercises 662 14.2.5 At-Risk-Set 663 14.3 Exploratory Data Analysis 663 14.3.1 A Complete Dataset 664 14.3.2 Sample Metrics 665 14.3.3 Histogram 669 14.3.4 Density Plot 670 14.3.5 Empirical Survivor Function 671 14.3.6 Q–Q Plot 673 14.4 Parameter Estimation 674 14.4.1 Estimators and Estimates 675 14.4.2 Properties of Estimators 675 14.4.3 Method of Moments Estimation 677 14.4.4 Maximum Likelihood Estimation 680 14.4.5 Exponentially Distributed Lifetimes 686 14.4.6 Weibull Distributed Lifetimes 692 14.5 The Kaplan–Meier Estimate 696 14.5.1 Motivation for the Kaplan–Meier Estimate Based a Complete Dataset 696 14.5.2 The Kaplan–Meier Estimator for a Censored Dataset 697 14.6 Cumulative Failure Rate Plots 701 14.6.1 The Nelson–Aalen Estimate of the Cumulative Failure Rate 703 14.7 Total-Time-on-Test Plotting 708 14.7.1 Total-Time-on-Test Plot for Complete Datasets 708 14.7.2 Total-Time-on-Test Plot for Censored Datasets 721 14.7.3 A Brief Comparison 722 14.8 Survival Analysis with Covariates 723 14.8.1 Proportional Hazards Model 723 14.8.2 Cox Models 726 14.8.3 Estimating the Parameters of the Cox Model 727 14.9 Problems 730 References 736 15 Bayesian Reliability Analysis 739 15.1 Introduction 739 15.1.1 Three Interpretations of Probability 739 15.1.2 Bayes’ Formula 741 15.2 Bayesian Data Analysis 742 15.2.1 Frequentist Data Analysis 743 15.2.2 Bayesian Data Analysis 743 15.2.3 Model for Observed Data 745 15.2.4 Prior Distribution 745 15.2.5 Observed Data 746 15.2.6 Likelihood Function 746 15.2.7 Posterior Distribution 747 15.3 Selection of Prior Distribution 749 15.3.1 Binomial Model 749 15.3.2 Exponential Model – Single Observation 752 15.3.3 Exponential Model – Multiple Observations 753 15.3.4 Homogeneous Poisson Process 755 15.3.5 Noninformative Prior Distributions 757 15.4 Bayesian Estimation 758 15.4.1 Bayesian Point Estimation 758 15.4.2 Credible Intervals 760 15.5 Predictive Distribution 761 15.6 Models with Multiple Parameters 762 15.7 Bayesian Analysis with R 762 15.8 Problems 764 References 766 16 Reliability Data: Sources and Quality 767 16.1 Introduction 767 16.1.1 Categories of Input Data 767 16.1.2 Parameters Estimates 768 16.2 Generic Reliability Databases 769 16.2.1 OREDA 770 16.2.2 PDS Data Handbook 772 16.2.3 PERD 773 16.2.4 SERH 773 16.2.5 NPRD, EPRD, and FMD 773 16.2.6 GADS 774 16.2.7 GIDEP 774 16.2.8 FMEDA Approach 775 16.2.9 Failure Event Databases 775 16.3 Reliability Prediction 775 16.3.1 MIL-HDBK-217 Approach 776 16.3.2 Similar Methods 778 16.4 Common Cause Failure Data 778 16.4.1 ICDE 779 16.4.2 IEC 61508 Method 779 16.5 Data Analysis and Data Quality 780 16.5.1 Outdated Technology 780 16.5.2 Inventory Data 781 16.5.3 Constant Failure Rates 781 16.5.4 Multiple Samples 783 16.5.5 Data From Manufacturers 785 16.5.6 Questioning the Data Quality 785 16.6 Data Dossier 785 16.6.1 Final Remarks 785 References 787 Appendix A Acronyms 789 Appendix B Laplace Transforms 793 B.1 Important Properties of Laplace Transforms 794 B.2 Laplace Transforms of Some Selected Functions 794 Author Index 797 Subject Index 803

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Book SynopsisIntroduces risk assessment with key theories, proven methods, and state-of-the-art applications Risk Assessment: Theory, Methods, and Applicationsremains one of the few textbooks to address current risk analysis and risk assessment with an emphasis on the possibility of sudden, major accidents across various areas of practicefrom machinery and manufacturing processes to nuclear power plants and transportation systems. Updated to align with ISO 31000 and other amended standards, this all-new2nd Editiondiscusses the main ideas and techniques for assessing risk today. The book begins with an introduction of risk analysis, assessment, and management, and includes a new section on the history of risk analysis. It covers hazards and threats, how to measure and evaluate risk, and risk management. It also adds new sections on risk governance and risk-informed decision making; combining accident theories and criteria for evaluating data sources; and subjectTable of ContentsPreface xiii Acknowledgments xvii About the Companion Site xix 1 Introduction 1 1.1 Risk in Our Modern Society 1 1.2 Important Trends 2 1.3 Major Accidents 4 1.4 History of Risk Assessment 4 1.5 Applications of Risk Assessment 9 1.6 Objectives, Scope, and Delimitation 11 1.7 Problems 12 References 13 2 The Words of Risk Analysis 15 2.1 Introduction 15 2.2 Risk 16 2.3 What Can Go Wrong? 20 2.4 What is the Likelihood? 38 2.5 What are the Consequences? 44 2.6 Additional Terms 49 2.7 Problems 54 References 56 3 Main Elements of Risk Assessment 59 3.1 Introduction 59 3.2 Risk Assessment Process 60 3.3 Risk Assessment Report 76 3.4 Risk Assessment in Safety Legislation 81 3.5 Validity and Quality Aspects of a Risk Assessment 82 3.6 Problems 83 References 84 4 Study Object and Limitations 87 4.1 Introduction 87 4.2 Study Object 87 4.3 Operating Context 91 4.4 System Modeling and Analysis 92 4.5 Complexity 95 4.6 Problems 97 References 98 5 Risk Acceptance 99 5.1 Introduction 99 5.2 Risk Acceptance Criteria 99 5.3 Approaches to Establishing Risk Acceptance Criteria 106 5.4 Risk Acceptance Criteria for Other Assets than Humans 114 5.5 Closure 115 5.6 Problems 115 References 117 6 Measuring Risk 121 6.1 Introduction 121 6.2 Risk Metrics 121 6.3 Measuring Risk to People 123 6.4 Risk Matrices 148 6.5 Reduction in Life Expectancy 154 6.6 Choice and Use of Risk Metrics 156 6.7 Risk Metrics for Other Assets 158 6.8 Problems 159 References 163 7 Risk Management 167 7.1 Introduction 167 7.2 Scope, Context, and Criteria 170 7.3 Risk Assessment 170 7.4 Risk Treatment 171 7.5 Communication and Consultation 172 7.6 Monitoring and Review 173 7.7 Recording and Reporting 174 7.8 Stakeholders 175 7.9 Risk and Decision-Making 176 7.10 Safety Legislation 179 7.11 Problems 180 References 180 8 Accident Models 183 8.1 Introduction 183 8.2 Accident Classification 183 8.3 Accident Investigation 188 8.4 Accident Causation 188 8.5 Accident Models 190 8.6 Energy and Barrier Models 193 8.7 Sequential Accident Models 195 8.8 Epidemiological Accident Models 201 8.9 Event Causation and Sequencing Models 208 8.10 Systemic Accident Models 213 8.11 Combining Accident Models 228 8.12 Problems 229 References 230 9 Data for Risk Analysis 235 9.1 Types of Data 235 9.2 Quality and Applicability of Data 238 9.3 Data Sources 239 9.4 Expert Judgment 250 9.5 Data Dossier 254 9.6 Problems 254 References 257 10 Hazard Identification 259 10.1 Introduction 259 10.2 Checklist Methods 263 10.3 Preliminary Hazard Analysis 266 10.4 Job Safety Analysis 278 10.5 FMECA 287 10.6 HAZOP 295 10.7 STPA 306 10.8 SWIFT 316 10.9 Comparing Semiquantitative Methods 322 10.10 Master Logic Diagram 322 10.11 Change Analysis 324 10.12 Hazard Log 327 10.13 Problems 331 References 335 11 Causal and Frequency Analysis 339 11.1 Introduction 339 11.2 Cause and Effect Diagram Analysis 341 11.3 Fault Tree Analysis 344 11.4 Bayesian Networks 370 11.5 Markov Methods 384 11.6 Problems 396 References 400 12 Development of Accident Scenarios 401 12.1 Introduction 401 12.2 Event Tree Analysis 402 12.3 Event Sequence Diagrams 426 12.4 Cause–Consequence Analysis 426 12.5 Hybrid Causal Logic 428 12.6 Escalation Problems 429 12.7 Consequence Models 429 12.8 Problems 431 References 435 13 Dependent Failures and Events 437 13.1 Introduction 437 13.2 Dependent Failures and Events 437 13.3 Dependency in Accident Scenarios 439 13.4 Cascading Failures 441 13.5 Common-Cause Failures 442 13.6 𝛽-Factor Model 452 13.7 Binomial Failure Rate Model 456 13.8 Multiple Greek Letter Model 457 13.9 𝛼-Factor Model 459 13.10 Multiple 𝛽-Factor Model 461 13.11 Problems 461 References 462 14 Barriers and Barrier Analysis 465 14.1 Introduction 465 14.2 Barriers and Barrier Classification 466 14.3 Barrier Management 474 14.4 Barrier Properties 476 14.5 Safety-Instrumented Systems 477 14.6 Hazard–Barrier Matrices 487 14.7 Safety Barrier Diagrams 488 14.8 Bow-Tie Diagrams 490 14.9 Energy Flow/Barrier Analysis 490 14.10 Layer of Protection Analysis 493 14.11 Barrier and Operational Risk Analysis 502 14.12 Systematic Identification and Evaluation of Risk Reduction Measures 512 14.13 Problems 518 References 520 15 Human Reliability Analysis 525 15.1 Introduction 525 15.2 Task Analysis 536 15.3 Human Error Identification 543 15.4 HRA Methods 552 15.5 Problems 573 References 574 16 Risk Analysis and Management for Operation 579 16.1 Introduction 579 16.2 Decisions About Risk 581 16.3 Aspects of Risk to Consider 583 16.4 Risk Indicators 585 16.5 Risk Modeling 594 16.6 Operational Risk Analysis – Updating the QRA 596 16.7 MIRMAP 598 16.8 Problems 601 References 602 17 Security Assessment 605 17.1 Introduction 605 17.2 Main Elements of Security Assessment 608 17.3 Industrial Control and Safety Systems 615 17.4 Security Assessment 617 17.5 Security Assessment Methods 625 17.6 Application Areas 626 17.7 Problems 627 References 628 18 Life Cycle Use of Risk Analysis 631 18.1 Introduction 631 18.2 Phases in the Life Cycle 631 18.3 Comments Applicable to all Phases 634 18.4 Feasibility and Concept Selection 635 18.5 Preliminary Design 637 18.6 Detailed Design and Construction 639 18.7 Operation and Maintenance 641 18.8 Major Modifications 641 18.9 Decommissioning and Removal 643 18.10 Problems 643 References 643 19 Uncertainty and Sensitivity Analysis 645 19.1 Introduction 645 19.2 Uncertainty 647 19.3 Categories of Uncertainty 648 19.4 Contributors to Uncertainty 651 19.5 Uncertainty Propagation 656 19.6 Sensitivity Analysis 661 19.7 Problems 663 References 664 20 Development and Applications of Risk Assessment 667 20.1 Introduction 667 20.2 Defense and Defense Industry 668 20.3 Nuclear Power Industry 670 20.4 Process Industry 674 20.5 Offshore Oil and Gas Industry 678 20.6 Space Industry 681 20.7 Aviation 683 20.8 Railway Transport 685 20.9 Marine Transport 686 20.10 Machinery Systems 689 20.11 Food Safety 690 20.12 Other Application Areas 692 20.13 Closure 695 References 697 Appendix A Elements of Probability Theory 701 A.1 Introduction 701 A.2 Outcomes and Events 701 A.3 Probability 706 A.4 Random Variables 710 A.5 Some Specific Distributions 718 A.6 Point and Interval Estimation 728 A.7 Bayesian Approach 732 A.8 Probability of Frequency Approach 733 References 739 Appendix B Acronyms 741 Author Index 747 Subject Index 753

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Book SynopsisTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started with Pre-Calculus 5 Chapter 1: Pre-Pre-Calculus 7 Pre-Calculus: An Overview 8 All the Number Basics (No, Not How to Count Them!) 9 The multitude of number types: Terms to know 9 The fundamental operations you can perform on numbers 11 The properties of numbers: Truths to remember 11 Visual Statements: When Math Follows Form with Function 12 Basic terms and concepts 13 Graphing linear equalities and inequalities 14 Gathering information from graphs 15 Get Yourself a Graphing Calculator 16 Chapter 2: Playing with Real Numbers 19 Solving Inequalities 19 Recapping inequality how-tos 20 Solving equations and inequalities when absolute value is involved 20 Expressing solutions for inequalities with interval notation 22 Variations on Dividing and Multiplying: Working with Radicals and Exponents 24 Defining and relating radicals and exponents 24 Rewriting radicals as exponents (or, creating rational exponents) 25 Getting a radical out of a denominator: Rationalizing 26 Chapter 3: The Building Blocks of Pre-Calculus Functions 31 Qualities of Special Function Types and Their Graphs 32 Even and odd functions 32 One-to-one functions 32 Dealing with Parent Functions and Their Graphs 33 Linear functions 33 Quadratic functions 33 Square-root functions 34 Absolute-value functions 34 Cubic functions 35 Cube-root functions 36 Graphing Functions That Have More Than One Rule: Piece-Wise Functions 37 Setting the Stage for Rational Functions 38 Step 1: Search for vertical asymptotes 39 Step 2: Look for horizontal asymptotes 40 Step 3: Seek out oblique asymptotes 41 Step 4: Locate the x- and y-intercepts 42 Putting the Results to Work: Graphing Rational Functions 42 Chapter 4: Operating on Functions 49 Transforming the Parent Graphs 50 Stretching and flattening 50 Translations 52 Reflections 54 Combining various transformations (a transformation in itself!) 55 Transforming functions point by point 57 Sharpen Your Scalpel: Operating on Functions 58 Adding and subtracting 59 Multiplying and dividing 60 Breaking down a composition of functions 60 Adjusting the domain and range of combined functions (if applicable) 61 Turning Inside Out with Inverse Functions 63 Graphing an inverse 64 Inverting a function to find its inverse 66 Verifying an inverse 66 Chapter 5: Digging Out and Using Roots to Graph Polynomial Functions 69 Understanding Degrees and Roots 70 Factoring a Polynomial Expression 71 Always the first step: Looking for a GCF 72 Unwrapping the box containing a trinomial 73 Recognizing and factoring special polynomials 74 Grouping to factor four or more terms 77 Finding the Roots of a Factored Equation 78 Cracking a Quadratic Equation When It Won’t Factor 79 Using the quadratic formula 79 Completing the square 80 Solving Unfactorable Polynomials with a Degree Higher Than Two 81 Counting a polynomial’s total roots 82 Tallying the real roots: Descartes’s rule of signs 82 Accounting for imaginary roots: The fundamental theorem of algebra 83 Guessing and checking the real roots 84 Put It in Reverse: Using Solutions to Find Factors 90 Graphing Polynomials 91 When all the roots are real numbers 91 When roots are imaginary numbers: Combining all techniques 95 Chapter 6: Exponential and Logarithmic Functions 97 Exploring Exponential Functions 98 Searching the ins and outs of exponential functions 98 Graphing and transforming exponential functions 100 Logarithms: The Inverse of Exponential Functions 102 Getting a better handle on logarithms 102 Managing the properties and identities of logs 103 Changing a log’s base 105 Calculating a number when you know its log: Inverse logs 105 Graphing logs 106 Base Jumping to Simplify and Solve Equations 109 Stepping through the process of exponential equation solving 109 Solving logarithmic equations 112 Growing Exponentially: Word Problems in the Kitchen 113 Part 2: The Essentials of Trigonometry 117 Chapter 7: Circling in on Angles 119 Introducing Radians: Circles Weren’t Always Measured in Degrees 120 Trig Ratios: Taking Right Triangles a Step Further 121 Making a sine 121 Looking for a cosine 122 Going on a tangent 124 Discovering the flip side: Reciprocal trig functions 125 Working in reverse: Inverse trig functions 126 Understanding How Trig Ratios Work on the Coordinate Plane 127 Building the Unit Circle by Dissecting the Right Way 129 Familiarizing yourself with the most common angles 129 Drawing uncommon angles 131 Digesting Special Triangle Ratios 132 The 45er: 45 -45 -90 triangle 132 The old 30-60: 30 -60 -90 triangle 133 Triangles and the Unit Circle: Working Together for the Common Good 135 Placing the major angles correctly, sans protractor 135 Retrieving trig-function values on the unit circle 138 Finding the reference angle to solve for angles on the unit circle 142 Measuring Arcs: When the Circle Is Put in Motion 146 Chapter 8: Simplifying the Graphing and Transformation of Trig Functions 149 Drafting the Sine and Cosine Parent Graphs 150 Sketching sine 150 Looking at cosine 152 Graphing Tangent and Cotangent 154 Tackling tangent 154 Clarifying cotangent 157 Putting Secant and Cosecant in Pictures 159 Graphing secant 159 Checking out cosecant 161 Transforming Trig Graphs 162 Messing with sine and cosine graphs 163 Tweaking tangent and cotangent graphs 173 Transforming the graphs of secant and cosecant 176 Chapter 9: Identifying with Trig Identities: The Basics 181 Keeping the End in Mind: A Quick Primer on Identities 182 Lining Up the Means to the End: Basic Trig Identities 182 Reciprocal and ratio identities 183 Pythagorean identities 185 Even/odd identities 188 Co-function identities 190 Periodicity identities 192 Tackling Difficult Trig Proofs: Some Techniques to Know 194 Dealing with demanding denominators 195 Going solo on each side 199 Chapter 10: Advanced Identities: Your Keys to Success 201 Finding Trig Functions of Sums and Differences 202 Searching out the sine of a b 202 Calculating the cosine of a b 206 Taming the tangent of a b 209 Doubling an Angle and Finding Its Trig Value 211 Finding the sine of a doubled angle 212 Calculating cosines for two 213 Squaring your cares away 215 Having twice the fun with tangents 216 Taking Trig Functions of Common Angles Divided in Two 217 A Glimpse of Calculus: Traveling from Products to Sums and Back 219 Expressing products as sums (or differences) 219 Transporting from sums (or differences) to products 220 Eliminating Exponents with Power-Reducing Formulas 221 Chapter 11: Taking Charge of Oblique Triangles with the Laws of Sines and Cosines 223 Solving a Triangle with the Law of Sines 224 When you know two angle measures 225 When you know two consecutive side lengths 228 Conquering a Triangle with the Law of Cosines 235 SSS: Finding angles using only sides 236 SAS: Tagging the angle in the middle (and the two sides) 238 Filling in the Triangle by Calculating Area 240 Finding area with two sides and an included angle (for SAS scenarios) 241 Using Heron’s Formula (for SSS scenarios) 241 Part 3: Analytic Geometry and System Solving 243 Chapter 12: Plane Thinking: Complex Numbers and Polar Coordinates 245 Understanding Real versus Imaginary 246 Combining Real and Imaginary: The Complex Number System 247 Grasping the usefulness of complex numbers 247 Performing operations with complex numbers 248 Graphing Complex Numbers 250 Plotting Around a Pole: Polar Coordinates 251 Wrapping your brain around the polar coordinate plane 252 Graphing polar coordinates with negative values 254 Changing to and from polar coordinates 256 Picturing polar equations 259 Chapter 13: Creating Conics by Slicing Cones 263 Cone to Cone: Identifying the Four Conic Sections 264 In picture (graph form) 264 In print (equation form) 266 Going Round and Round: Graphing Circles 267 Graphing circles at the origin 267 Graphing circles away from the origin 268 Writing in center–radius form 269 Riding the Ups and Downs with Parabolas 270 Labeling the parts 270 Understanding the characteristics of a standard parabola 271 Plotting the variations: Parabolas all over the plane 272 The vertex, axis of symmetry, focus, and directrix 273 Identifying the min and max of vertical parabolas 276 The Fat and the Skinny on the Ellipse 278 Labeling ellipses and expressing them with algebra 279 Identifying the parts from the equation 281 Pair Two Curves and What Do You Get? Hyperbolas 284 Visualizing the two types of hyperbolas and their bits and pieces 284 Graphing a hyperbola from an equation 287 Finding the equations of asymptotes 287 Expressing Conics Outside the Realm of Cartesian Coordinates 289 Graphing conic sections in parametric form 290 The equations of conic sections on the polar coordinate plane 292 Chapter 14: Streamlining Systems, Managing Variables 295 A Primer on Your System-Solving Options 296 Algebraic Solutions of Two-Equation Systems 297 Solving linear systems 297 Working nonlinear systems 300 Solving Systems with More than Two Equations 304 Decomposing Partial Fractions 306 Surveying Systems of Inequalities 307 Introducing Matrices: The Basics 309 Applying basic operations to matrices 310 Multiplying matrices by each other 311 Simplifying Matrices to Ease the Solving Process 312 Writing a system in matrix form 313 Reduced row-echelon form 313 Augmented form 314 Making Matrices Work for You 315 Using Gaussian elimination to solve systems 316 Multiplying a matrix by its inverse 320 Using determinants: Cramer’s Rule 323 Chapter 15: Sequences, Series, and Expanding Binomials for the Real World 327 Speaking Sequentially: Grasping the General Method 328 Determining a sequence’s terms 328 Working in reverse: Forming an expression from terms 329 Recursive sequences: One type of general sequence 330 Difference between Terms: Arithmetic Sequences 331 Using consecutive terms to find another 332 Using any two terms 332 Ratios and Consecutive Paired Terms: Geometric Sequences 334 Identifying a particular term when given consecutive terms 334 Going out of order: Dealing with nonconsecutive terms 335 Creating a Series: Summing Terms of a Sequence 337 Reviewing general summation notation 337 Summing an arithmetic sequence 338 Seeing how a geometric sequence adds up 339 Expanding with the Binomial Theorem 342 Breaking down the binomial theorem 344 Expanding by using the binomial theorem 345 Chapter 16: Onward to Calculus 351 Scoping Out the Differences between Pre-Calculus and Calculus 352 Understanding Your Limits 353 Finding the Limit of a Function 355 Graphically 355 Analytically 356 Algebraically 357 Operating on Limits: The Limit Laws 361 Calculating the Average Rate of Change 362 Exploring Continuity in Functions 363 Determining whether a function is continuous 364 Discontinuity in rational functions 365 Part 4: The Part of Tens 367 Chapter 17: Ten Polar Graphs 369 Spiraling Outward 369 Falling in Love with a Cardioid 370 Cardioids and Lima Beans 370 Leaning Lemniscates 371 Lacing through Lemniscates 372 Roses with Even Petals 372 A rose Is a Rose Is a Rose 373 Limaçon or Escargot? 373 Limaçon on the Side 374 Bifolium or Rabbit Ears? 374 Chapter 18: Ten Habits to Adjust before Calculus 375 Figure Out What the Problem Is Asking 375 Draw Pictures (the More the Better) 376 Plan Your Attack — Identify Your Targets 377 Write Down Any Formulas 377 Show Each Step of Your Work 378 Know When to Quit 378 Check Your Answers 379 Practice Plenty of Problems 380 Keep Track of the Order of Operations 380 Use Caution When Dealing with Fractions 381 Index 383

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Book SynopsisGet a handle on pre-calculus in a pinch! If you're tackling pre-calculus and want to up your chances of doing your very best, this hands-on workbook is just what you need to grasp and retain the concepts that will help you succeed. Inside, you'll get basic content review for every concept, paired with examples and plenty of practice problems, ample workspace, step-by-step solutions, and thorough explanations for each and every problem. In Pre-Calculus Workbook For Dummies, you'll also get free access to a quiz for every chapter online! With all of the lessons and practice offered, you'll memorize the most frequently used formulas, see how to avoid common mistakes, understand tricky trig proofs, and get the inside scoop on key concepts such as quadratic equations. Get ample review before jumping into a calculus course Supplement your classroom work with easy-to-follow guidance Make complex formulas and concepts more approachaTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Setting the Foundation: The Nuts And Bolts of Pre-Calculus 5 Chapter 1: Preparing for Pre-Calculus 7 Reviewing Order of Operations: The Fun in Fundamentals 8 Keeping Your Balance While Solving Equalities 10 When Your Image Really Counts: Graphing Equalities and Inequalities 12 Graphing with two points 12 Graphing by using the slope-intercept form 13 Graphing inequalities 14 Using Graphs to Find Distance, Midpoint, and Slope 15 Finding the distance 15 Calculating the midpoint 16 Discovering the slope 16 Answers to Problems on Fundamentals 19 Chapter 2: Real Numbers Come Clean 25 Solving Inequalities 25 Expressing Inequality Solutions in Interval Notation 28 Radicals and Exponents — Just Simplify! 30 Getting Out of a Sticky Situation, or Rationalizing 33 Answers to Problems on Real Numbers 35 Chapter 3: Controlling Functions by Knowing Their Function 39 Using Both Faces of the Coin: Even and Odd 40 Leaving the Nest: Transforming Parent Graphs 42 Quadratic functions 42 Square root functions 42 Absolute value functions 43 Cubic functions 43 Cube root functions 44 Steeper or flatter 44 Translations 46 Reflections 46 Combinations of transformations 46 Graphing Rational Functions 49 Piecing Together Piecewise Functions 52 Combining Functions 54 Evaluating Composition of Functions 55 Working Together: Domain and Range 57 Unlocking the Inverse of a Function: Turning It Inside Out 59 Answers to Problems on Functions 61 Chapter 4: Searching for Roots 75 Factoring a Factorable Quadratic 75 Solving a Quadratic Polynomial Equation 78 Completing the square 78 Quadratic formula 79 Solving High-Order Polynomials 80 Factoring by grouping 80 Determining positive and negative roots: Descartes’ Rule of Signs 81 Counting on imaginary roots 81 Getting the rational roots 81 Finding roots through synthetic division 82 Using Roots to Create an Equation 84 Graphing Polynomials 85 Answers to Problems on Roots and Degrees 89 Chapter 5: Exponential and Logarithmic Functions 95 Working with Exponential Functions 95 Eagerly Engaging Edgy Logarithmic Solutions 98 Making Exponents and Logs Work Together 101 Using Exponents and Logs in Practical Applications 103 Answers to Problems on Exponential and Logarithmic Functions 106 Part 2: Trig is the Key: Basic Review, The Unit Circle, and Graphs 113 Chapter 6: Basic Trigonometry and the Unit Circle 115 Finding the Six Trigonometric Ratios 115 Solving Word Problems with Right Triangles 118 Unit Circle and the Coordinate Plane: Finding Points and Angles 121 Finding Ratios from Angles on the Unit Circle 124 Solving Trig Equations 127 Making and Measuring Arcs 129 Answers to Problems on Basic Trig and the Unit Circle 131 Chapter 7: Graphing and Transforming Trig Functions 137 Getting a Grip on Periodic Graphs 137 Parent Graphs and Transformations: Sine and Cosine 138 Tangent and Cotangent: More Family Members 141 Generations: Secant and Cosecant 143 Answers to Problems on Graphing and Transforming Trig Functions 147 Part 3: Digging Into Advanced Trig: Identities, Theorems, and Applications 155 Chapter 8: Basic Trig Identities 157 Using Reciprocal Identities to Simplify Trig Expressions 157 Simplifying with Pythagorean Identities 159 Discovering Even-Odd Identities 160 Simplifying with Co-Function Identities 162 Moving with Periodicity Identities 163 Tackling Trig Proofs (Identities) 165 Answers to Problems on Basic Trig Identities 167 Chapter 9: Advanced Trig Identities 175 Simplifying with Sum and Difference Identities 175 Using Double-Angle Identities 178 Reducing with Half-Angle Identities 180 Changing Products to Sums 181 Expressing Sums as Products 182 Powering Down: Power-Reducing Formulas 184 Answers to Problems on Advanced Trig Identities 186 Chapter 10: Solving Oblique Triangles 193 Solving a Triangle with the Law of Sines: ASA and AAS 194 Tackling Triangles in the Ambiguous Case: SSA 195 Conquering a Triangle with the Law of Cosines: SAS and SSS 197 Using Oblique Triangles to Solve Practical Applications 198 Figuring Area 201 Answers to Problems on Solving Triangles 202 Part 4: Polar Coordinates, Cones, Solutions, Sequences, and Finding Your Limits 209 Chapter 11: Exploring Complex Numbers and Polar Coordinates 211 Performing Operations with and Graphing Complex Numbers 212 Round a Pole: Graphing Polar Coordinates 215 Changing to and from Polar 217 Graphing Polar Equations 220 Archimedean spiral 220 Cardioid 220 Rose 220 Circle 220 Lemniscate 220 Limaçon 221 Answers to Problems on Complex Numbers and Polar Coordinates 223 Chapter 12: Conquering Conic Sections 229 A Quick Conic Review 230 Going Round and Round with Circles 230 The Ups and Downs: Graphing Parabolas 232 Standing tall: Vertical parabolas 233 Lying down on the job: Horizontal parabolas 235 The Fat and the Skinny: Graphing Ellipses 237 Short and fat: The horizontal ellipse 237 Tall and skinny: The vertical ellipse 239 No Caffeine Required: Graphing Hyperbolas 241 Horizontal hyperbolas 241 Vertical hyperbolas 244 Identifying Conic Sections 246 Conic Sections in Parametric Form and Polar Coordinates 248 Parametric form for conic sections 248 Changing from parametric form to rectangular form 250 Conic sections on the polar coordinate plane 251 Answers to Problems on Conic Sections 253 Chapter 13: Finding Solutions for Systems of Equations 265 A Quick-and-Dirty Technique Overview 266 Solving Two Linear Equations with Two Variables 266 The substitution method 267 The elimination method 268 Not-So-Straight: Solving Nonlinear Systems 269 One equation that’s linear and one that isn’t 269 Two nonlinear equations 270 Systems of rational equations 271 Systems of More Than Two Equations 272 Graphing Systems of Inequalities 274 Breaking Down Decomposing Partial Fractions 276 Working with a Matrix 278 Getting It in the Right Form: Simplifying Matrices 281 Solving Systems of Equations Using Matrices 283 Gaussian elimination 283 Inverse matrices 285 Cramer’s Rule 287 Answers to Problems on Systems of Equations 289 Chapter 14: Spotting Patterns in Sequences and Series 301 General Sequences and Series: Determining Terms 301 Working Out the Common Difference: Arithmetic Sequences and Series 303 Simplifying Geometric Sequences and Series 305 Expanding Polynomials Using the Binomial Theorem 308 Answers to Problems on Sequences, Series, and Binomials 310 Chapter 15: Previewing Calculus 315 Finding Limits: Graphically, Analytically, and Algebraically 316 Graphically 316 Analytically 318 Algebraically 319 Knowing Your Limits 321 Calculating the Average Rate of Change 322 Determining Continuity 323 Answers to Problems on Calculus 326 Part 5: The Part of Tens 329 Chapter 16: Ten Plus Parent Graphs 331 Squaring Up with Quadratics 331 Cueing Up for Cubics 332 Rooting for Square Roots and Cube Roots 333 Graphing Absolutely Fabulous Absolute Value Functions 334 Flipping over Rational Functions 334 Exploring Exponential Graphs and Logarithmic Graphs 335 Seeing the Sine and Cosine 336 Covering Cosecant and Secant 337 Tripping over Tangent and Cotangent 338 Lining Up and Going Straight with Lines 339 Chapter 17: Ten Missteps to Avoid 341 Going Out of Order (of Operations) 341 FOILing Binomials Incorrectly 342 Breaking Up Fractions Badly 342 Combining Terms That Can’t Be Combined 342 Forgetting to Flip the Fraction 342 Losing the Negative (Sign) 343 Oversimplifying Roots 343 Executing Exponent Errors 343 Ignoring Extraneous 344 Misinterpreting Trig Notation 344 Index 345

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Book Synopsis

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Book SynopsisTable of ContentsPreface xiii Part One — Matrices 1 Basic properties of vectors and matrices 3 1 Introduction 3 2 Sets 3 3 Matrices: addition and multiplication 4 4 The transpose of a matrix 6 5 Square matrices 6 6 Linear forms and quadratic forms 7 7 The rank of a matrix 9 8 The inverse 10 9 The determinant 10 10 The trace 11 11 Partitioned matrices 12 12 Complex matrices 14 13 Eigenvalues and eigenvectors 14 14 Schur’s decomposition theorem 17 15 The Jordan decomposition 18 16 The singular-value decomposition 20 17 Further results concerning eigenvalues 20 18 Positive (semi)definite matrices 23 19 Three further results for positive definite matrices 25 20 A useful result 26 21 Symmetric matrix functions 27 Miscellaneous exercises 28 Bibliographical notes 30 2 Kronecker products, vec operator, and Moore-Penrose inverse 31 1 Introduction 31 2 The Kronecker product 31 3 Eigenvalues of a Kronecker product 33 4 The vec operator 34 5 The Moore-Penrose (MP) inverse 36 6 Existence and uniqueness of the MP inverse 37 7 Some properties of the MP inverse 38 8 Further properties 39 9 The solution of linear equation systems 41 Miscellaneous exercises 43 Bibliographical notes 45 3 Miscellaneous matrix results 47 1 Introduction 47 2 The adjoint matrix 47 3 Proof of Theorem 3.1 49 4 Bordered determinants 51 5 The matrix equation AX = 0 51 6 The Hadamard product 52 7 The commutation matrix Kmn 54 8 The duplication matrix Dn 56 9 Relationship between Dn+1 and Dn, I 58 10 Relationship between Dn+1 and Dn, II 59 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) 63 13 The bordered Gramian matrix 65 14 The equations X1A + X2B′ = G1,X1B = G2 67 Miscellaneous exercises 69 Bibliographical notes 70 Part Two — Differentials: the theory 4 Mathematical preliminaries 73 1 Introduction 73 2 Interior points and accumulation points 73 3 Open and closed sets 75 4 The Bolzano-Weierstrass theorem 77 5 Functions 78 6 The limit of a function 79 7 Continuous functions and compactness 80 8 Convex sets 81 9 Convex and concave functions 83 Bibliographical notes 86 5 Differentials and differentiability 87 1 Introduction 87 2 Continuity 88 3 Differentiability and linear approximation 90 4 The differential of a vector function 91 5 Uniqueness of the differential 93 6 Continuity of differentiable functions 94 7 Partial derivatives 95 8 The first identification theorem 96 9 Existence of the differential, I 97 10 Existence of the differential, II 99 11 Continuous differentiability 100 12 The chain rule 100 13 Cauchy invariance 102 14 The mean-value theorem for real-valued functions 103 15 Differentiable matrix functions 104 16 Some remarks on notation 106 17 Complex differentiation 108 Miscellaneous exercises 110 Bibliographical notes 110 6 The second differential 111 1 Introduction 111 2 Second-order partial derivatives 111 3 The Hessian matrix 112 4 Twice differentiability and second-order approximation, I 113 5 Definition of twice differentiability 114 6 The second differential 115 7 Symmetry of the Hessian matrix 117 8 The second identification theorem 119 9 Twice differentiability and second-order approximation, II 119 10 Chain rule for Hessian matrices 121 11 The analog for second differentials 123 12 Taylor’s theorem for real-valued functions 124 13 Higher-order differentials 125 14 Real analytic functions 125 15 Twice differentiable matrix functions 126 Bibliographical notes 127 7 Static optimization 129 1 Introduction 129 2 Unconstrained optimization 130 3 The existence of absolute extrema 131 4 Necessary conditions for a local minimum 132 5 Sufficient conditions for a local minimum: first-derivative test 134 6 Sufficient conditions for a local minimum: second-derivative test 136 7 Characterization of differentiable convex functions 138 8 Characterization of twice differentiable convex functions 141 9 Sufficient conditions for an absolute minimum 142 10 Monotonic transformations 143 11 Optimization subject to constraints 144 12 Necessary conditions for a local minimum under constraints 145 13 Sufficient conditions for a local minimum under constraints 149 14 Sufficient conditions for an absolute minimum under constraints 154 15 A note on constraints in matrix form 155 16 Economic interpretation of Lagrange multipliers 155 Appendix: the implicit function theorem 157 Bibliographical notes 159 Part Three — Differentials: the practice 8 Some important differentials 163 1 Introduction 163 2 Fundamental rules of differential calculus 163 3 The differential of a determinant 165 4 The differential of an inverse 168 5 Differential of the Moore-Penrose inverse 169 6 The differential of the adjoint matrix 172 7 On differentiating eigenvalues and eigenvectors 174 8 The continuity of eigenprojections 176 9 The differential of eigenvalues and eigenvectors: symmetric case 180 10 Two alternative expressions for dλ 183 11 Second differential of the eigenvalue function 185 Miscellaneous exercises 186 Bibliographical notes 189 9 First-order differentials and Jacobian matrices 191 1 Introduction 191 2 Classification 192 3 Derisatives 192 4 Derivatives 194 5 Identification of Jacobian matrices 196 6 The first identification table 197 7 Partitioning of the derivative 197 8 Scalar functions of a scalar 198 9 Scalar functions of a vector 198 10 Scalar functions of a matrix, I: trace 199 11 Scalar functions of a matrix, II: determinant 201 12 Scalar functions of a matrix, III: eigenvalue 202 13 Two examples of vector functions 203 14 Matrix functions 204 15 Kronecker products 206 16 Some other problems 208 17 Jacobians of transformations 209 Bibliographical notes 210 10 Second-order differentials and Hessian matrices 211 1 Introduction 211 2 The second identification table 211 3 Linear and quadratic forms 212 4 A useful theorem 213 5 The determinant function 214 6 The eigenvalue function 215 7 Other examples 215 8 Composite functions 217 9 The eigenvector function 218 10 Hessian of matrix functions, I 219 11 Hessian of matrix functions, II 219 Miscellaneous exercises 220 Part Four — Inequalities 11 Inequalities 225 1 Introduction 225 2 The Cauchy-Schwarz inequality 226 3 Matrix analogs of the Cauchy-Schwarz inequality 227 4 The theorem of the arithmetic and geometric means 228 5 The Rayleigh quotient 230 6 Concavity of λ1 and convexity of λn 232 7 Variational description of eigenvalues 232 8 Fischer’s min-max theorem 234 9 Monotonicity of the eigenvalues 236 10 The Poincar´e separation theorem 236 11 Two corollaries of Poincar´e’s theorem 237 12 Further consequences of the Poincar´e theorem 238 13 Multiplicative version 239 14 The maximum of a bilinear form 241 15 Hadamard’s inequality 242 16 An interlude: Karamata’s inequality 242 17 Karamata’s inequality and eigenvalues 244 18 An inequality concerning positive semidefinite matrices 245 19 A representation theorem for ( ∑api )1/p 246 20 A representation theorem for (trAp)1/p 247 21 Hölder’s inequality 248 22 Concavity of log|A| 250 23 Minkowski’s inequality 251 24 Quasilinear representation of |A|1/n 253 25 Minkowski’s determinant theorem 255 26 Weighted means of order p 256 27 Schlömilch’s inequality 258 28 Curvature properties of Mp(x, a) 259 29 Least squares 260 30 Generalized least squares 261 31 Restricted least squares 262 32 Restricted least squares: matrix version 264 Miscellaneous exercises 265 Bibliographical notes 269 Part Five — The linear model 12 Statistical preliminaries 273 1 Introduction 273 2 The cumulative distribution function 273 3 The joint density function 274 4 Expectations 274 5 Variance and covariance 275 6 Independence of two random variables 277 7 Independence of n random variables 279 8 Sampling 279 9 The one-dimensional normal distribution 279 10 The multivariate normal distribution 280 11 Estimation 282 Miscellaneous exercises 282 Bibliographical notes 283 13 The linear regression model 285 1 Introduction 285 2 Affine minimum-trace unbiased estimation 286 3 The Gauss-Markov theorem 287 4 The method of least squares 290 5 Aitken’s theorem 291 6 Multicollinearity 293 7 Estimable functions 295 8 Linear constraints: the case M(R′) ⊂M(X′) 296 9 Linear constraints: the general case 300 10 Linear constraints: the case M(R′) ∩M(X′) = {0} 302 11 A singular variance matrix: the case M(X) ⊂M(V ) 304 12 A singular variance matrix: the case r(X′V +X) = r(X) 305 13 A singular variance matrix: the general case, I 307 14 Explicit and implicit linear constraints 307 15 The general linear model, I 310 16 A singular variance matrix: the general case, II 311 17 The general linear model, II 314 18 Generalized least squares 315 19 Restricted least squares 316 Miscellaneous exercises 318 Bibliographical notes 319 14 Further topics in the linear model 321 1 Introduction 321 2 Best quadratic unbiased estimation of σ2 322 3 The best quadratic and positive unbiased estimator of σ2 322 4 The best quadratic unbiased estimator of σ2 324 5 Best quadratic invariant estimation of σ2 326 6 The best quadratic and positive invariant estimator of σ2 327 7 The best quadratic invariant estimator of σ2 329 8 Best quadratic unbiased estimation: multivariate normal case 330 9 Bounds for the bias of the least-squares estimator of σ2, I 332 10 Bounds for the bias of the least-squares estimator of σ2, II 333 11 The prediction of disturbances 335 12 Best linear unbiased predictors with scalar variance matrix 336 13 Best linear unbiased predictors with fixed variance matrix, I 338 14 Best linear unbiased predictors with fixed variance matrix, II 340 15 Local sensitivity of the posterior mean 341 16 Local sensitivity of the posterior precision 342 Bibliographical notes 344 Part Six — Applications to maximum likelihood estimation 15 Maximum likelihood estimation 347 1 Introduction 347 2 The method of maximum likelihood (ML) 347 3 ML estimation of the multivariate normal distribution 348 4 Symmetry: implicit versus explicit treatment 350 5 The treatment of positive definiteness 351 6 The information matrix 352 7 ML estimation of the multivariate normal distribution: distinct means 354 8 The multivariate linear regression model 354 9 The errors-in-variables model 357 10 The nonlinear regression model with normal errors 359 11 Special case: functional independence of mean and variance parameters 361 12 Generalization of Theorem 15.6 362 Miscellaneous exercises 364 Bibliographical notes 365 16 Simultaneous equations 367 1 Introduction 367 2 The simultaneous equations model 367 3 The identification problem 369 4 Identification with linear constraints on B and Γ only 371 5 Identification with linear constraints on B, Γ, and ∑ 371 6 Nonlinear constraints 373 7 FIML: the information matrix (general case) 374 8 FIML: asymptotic variance matrix (special case) 376 9 LIML: first-order conditions 378 10 LIML: information matrix 381 11 LIML: asymptotic variance matrix 383 Bibliographical notes 388 17 Topics in psychometrics 389 1 Introduction 389 2 Population principal components 390 3 Optimality of principal components 391 4 A related result 392 5 Sample principal components 393 6 Optimality of sample principal components 395 7 One-mode component analysis 395 8 One-mode component analysis and sample principal components 398 9 Two-mode component analysis 399 10 Multimode component analysis 400 11 Factor analysis 404 12 A zigzag routine 407 13 A Newton-Raphson routine 408 14 Kaiser’s varimax method 412 15 Canonical correlations and variates in the population 414 16 Correspondence analysis 417 17 Linear discriminant analysis 418 Bibliographical notes 419 Part Seven — Summary 18 Matrix calculus: the essentials 423 1 Introduction 423 2 Differentials 424 3 Vector calculus 426 4 Optimization 429 5 Least squares 431 6 Matrix calculus 432 7 Interlude on linear and quadratic forms 434 8 The second differential 434 9 Chain rule for second differentials 436 10 Four examples 438 11 The Kronecker product and vec operator 439 12 Identification 441 13 The commutation matrix 442 14 From second differential to Hessian 443 15 Symmetry and the duplication matrix 444 16 Maximum likelihood 445 Further reading 448 Bibliography 449 Index of symbols 467 Subject index 471

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Book SynopsisBoost your chances of scoring higher at Algebra II Algebra II introduces students to complex algebra concepts in preparation for trigonometry and calculus. In this new edition of Algebra II Workbook For Dummies, high school and college students will work through the types of Algebra II problems they''ll see in class, including systems of equations, matrices, graphs, and conic sections. Plus, the book now comes with free 1-year access to chapter quizzes online! A recent report by ACT shows that over a quarter of ACT-tested 2012 high school graduates did not meet any of the four college readiness benchmarks in mathematics, English, reading, and science. Algebra II Workbook For Dummies presents tricky topics in plain English and short lessons, with examples and practice at every step to help students master the essentials, setting them up for success with each new lesson. Tracks to a typical Algebra II class Can be used as a suppTable of ContentsIntroduction About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started With Algebra II 5 Chapter 1: Going Beyond Beginning Algebra 7 Good Citizenship: Following the Order of Operations and Other Properties 7 Specializing in Products and FOIL 10 Variables on the Side: Solving Linear Equations 11 Dealing with Linear Absolute Value Equations 12 Greater Math Skills: Equalizing Linear Inequalities 14 Answers to Problems on Going Beyond Beginning Algebra 16 Chapter 2: Handling Quadratic (and Quadratic-Like) Equations and Inequalities 21 Finding Reasonable Solutions with Radicals 22 UnFOILed Again! Successfully Factoring for Solutions 23 Your Bag of Tricks: Factoring Multiple Ways 25 Keeping Your Act Together: Factoring by Grouping 26 Resorting to the Quadratic Formula 27 Solving Quadratics by Completing the Square 29 Working with Quadratic-Like Equations 30 Checking Out Quadratic Inequalities 32 Answers to Problems on Quadratic (and Quadratic-Like) Equations and Inequalities 34 Chapter 3: Rooting Out the Rational, the Radical, and the Negative 43 Doing Away with Denominators with an LCD 44 Simplifying and Solving Proportions 46 Wrangling with Radicals 48 Changing Negative Attitudes toward Negative Exponents 49 Divided Powers: Solving Equations with Fractional Exponents 51 Answers to Problems on Rooting Out the Rational, the Radical, and the Negative 53 Chapter 4: Graphing for the Good Life 61 Coordinating Axes, Coordinates of Points, and Quadrants 62 Crossing the Line: Using Intercepts and Symmetry to Graph 64 Graphing Lines Using Slope-Intercept and Standard Forms 67 Graphing Basic Polynomial Curves 69 Grappling with Radical and Absolute Value Functions 71 Enter the Machines: Using a Graphing Calculator 73 Answers to Problems on Graphing for the Good Life 77 Part 2: Functions 89 Chapter 5: Formulating Functions 91 Evaluating Functions 91 Determining the Domain and Range of a Function 93 Recognizing Even, Odd, and One-to-One Functions 94 Composing Functions and Simplifying the Difference Quotient 96 Solving for Inverse Functions 99 Answers to Problems on Formulating Functions 101 Chapter 6: Specializing in Quadratic Functions 107 Finding Intercepts and the Vertex of a Parabola 108 Applying Quadratics to Real-Life Situations 109 Graphing Parabolas 111 Answers to Problems on Quadratic Functions 113 Chapter 7: Plugging in Polynomials 119 Finding Basic Polynomial Intercepts 120 Digging up More-Difficult Polynomial Roots with Factoring 122 Determining Where a Function Is Positive or Negative 123 Graphing Polynomials 125 Possible Roots and Where to Find Them: The Rational Root Theorem and Descartes’s Rule 127 Getting Real Results with Synthetic Division and the Remainder Theorem 130 Connecting the Factor Theorem with a Polynomial’s Roots 132 Answers to Problems on Plugging in Polynomials 134 Chapter 8: Acting Rationally with Functions 143 Determining Domain and Intercepts of Rational Functions 144 Introducing Vertical and Horizontal Asymptotes 145 Getting a New Slant with Oblique Asymptotes 147 Removing Discontinuities 148 Going the Limit: Limits at a Number and Infinity 149 Graphing Rational Functions 151 Answers to Problems on Rational Functions 156 Chapter 9: Exposing Exponential and Logarithmic Functions 163 Evaluating e-Expressions and Powers of e 164 Solving Exponential Equations 165 Making Cents: Applying Compound Interest and Continuous Compounding 167 Checking out the Properties of Logarithms 169 Presto-Chango: Expanding and Contracting Expressions with Log Functions 171 Solving Logarithmic Equations 173 They Ought to Be in Pictures: Graphing Exponential and Logarithmic Functions 175 Answers to Problems on Exponential and Logarithmic Functions 179 Part 3: Conics And Systems Of Equations 189 Chapter 10: Any Way You Slice It: Conic Sections 191 Putting Equations of Parabolas in Standard Form 192 Shaping Up: Determining the Focus and Directrix of a Parabola 194 Back to the Drawing Board: Sketching Parabolas 196 Writing the Equations of Circles and Ellipses in Standard Form 198 Determining Foci and Vertices of Ellipses 201 Rounding Out Your Sketches: Circles and Ellipses 203 Hyperbola: Standard Equations and Foci 205 Determining the Asymptotes and Intercepts of Hyperbolas 206 Sketching the Hyperbola 208 Answers to Problems on Conic Sections 211 Chapter 11: Solving Systems of Linear Equations 221 Solving Two Linear Equations Algebraically 221 Using Cramer’s Rule to Defeat Unruly Fractions 223 A Third Variable: Upping the Systems to Three Linear Equations 225 A Line by Any Other Name: Writing Generalized Solution Rules 227 Decomposing Fractions Using Systems 229 Answers to Problems on Systems of Equations 231 Chapter 12: Solving Systems of Nonlinear Equations and Inequalities 237 Finding the Intersections of Lines and Parabolas 237 Crossing Curves: Finding the Intersections of Parabolas and Circles 239 Appealing to a Higher Power: Dealing with Exponential Systems 240 Solving Systems of Inequalities 242 Answers to Problems on Solving Systems of Nonlinear Equations and Inequalities 245 Part 4: Other Good Stuff: Lists, Arrays, And Imaginary Numbers 251 Chapter 13: Getting More Complex with Imaginary Numbers 253 Simplifying Powers of i 254 Not Quite Brain Surgery: Doing Operations on Complex Numbers 255 “Dividing” Complex Numbers with a Conjugate 256 Solving Equations with Complex Solutions 257 Answers to Problems on Imaginary Numbers 259 Chapter 14: Getting Squared Away with Matrices 263 Describing Dimensions and Types of Matrices 263 Adding, Subtracting, and Doing Scalar Multiplication on Matrices 265 Trying Times: Multiplying Matrices by Each Other 267 The Search for Identity: Finding Inverse Matrices 268 Using Matrices to Solve Systems of Equations 272 Answers to Problems on Matrices 274 Chapter 15: Going Out of Sequence with Sequences and Series 279 Writing the Terms of a Sequence 279 Differences and Multipliers: Working with Special Sequences 282 Backtracking: Constructing Recursively Defined Sequences 283 Using Summation Notation 284 Finding Sums with Special Series 286 Answers to Problems on Sequences and Series 289 Chapter 16: Everything You Ever Wanted to Know about Sets and Counting 293 Writing the Elements of a Set from Rules or Patterns 294 Get Together: Combining Sets with Unions, Intersections, and Complements 295 Multiplication Countdowns: Simplifying Factorial Expressions 297 Checking Your Options: Using the Multiplication Property 298 Counting on Permutations When Order Matters 300 Mixing It Up with Combinations 301 Raising Binomials to Powers: Investigating the Binomial Theorem 303 Answers to Problems on Sets and Counting 304 Part 5: The Part Of Tens 309 Chapter 17: Basic Graphs 311 Putting Polynomials in Their Place 311 Lining Up Front and Center 312 Being Absolutely Sure with Absolute Value 313 Graphing Reciprocals of x and x2 .313 Rooting Out Square Root and Cube Root .314 Growing Exponentially with a Graph 315 Logging In on Logarithmic Graphing 316 Chapter 18: Ten Special Sequences and Their Sums 317 Adding as Easy as One, Two, Three 317 Summing Up the Squares 318 Finding the Sum of the Cubes 318 Not Being at Odds with Summing Odd Numbers 319 Evening Things Out by Adding Up Even Numbers 319 Adding Everything Arithmetic 319 Geometrically Speaking 320 Easing into a Sum for e 320 Signing In on the Sine 321 Powering Up on Powers of 2 322 Adding Up Fractions with Multiples for Denominators 322 Index 323

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Book SynopsisTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 4 Where to Go from Here 4 Part 1: Homing In On Basic Solutions 5 Chapter 1: Going Beyond Beginning Algebra 7 Outlining Algebraic Properties 8 Keeping order with the commutative property 8 Maintaining group harmony with the associative property 9 Distributing a wealth of values 9 Checking out an algebraic ID 10 Singing along in-verses 11 Ordering Your Operations 11 Zeroing in on the Multiplication Property of Zero 12 Expounding on Exponential Rules 13 Multiplying and dividing exponents 13 Getting to the roots of exponents 14 Raising or lowering the roof with exponents 14 Making nice with negative exponents 15 Implementing Factoring Techniques 15 Factoring two terms 16 Taking on three terms 17 Factoring four or more terms by grouping 19 Chapter 2: Toeing the Straight Line: Linear Equations 21 Linear Equations: Handling the First Degree 21 Tackling basic linear equations 22 Clearing out fractions 23 Isolating different unknowns 24 Linear Inequalities: Algebraic Relationship Therapy 25 Solving linear inequalities 26 Introducing interval notation 27 Compounding inequality issues 28 Absolute Value: Keeping Everything in Line 30 Solving absolute value equations 31 Seeing through absolute value inequality 31 Chapter 3: Conquering Quadratic Equations 35 Implementing the Square Root Rule 36 Dismantling Quadratic Equations into Factors 37 Factoring binomials 37 Factoring trinomials 39 Factoring by grouping 40 Resorting to the Quadratic Formula 41 Finding rational solutions 42 Straightening out irrational solutions 42 Formulating huge quadratic results 43 Completing the Square: Warming Up for Conics 43 Squaring up a quadratic equation 44 Completing the square twice over 45 Tackling Higher-Powered Polynomials 46 Handling the sum or difference of cubes 47 Tackling quadratic-like trinomials 48 Solving Quadratic Inequalities 49 Keeping inequality strictly quadratic 50 Signing up for fractions 52 Increasing the number of factors 53 Considering absolute value inequalities 53 Chapter 4: Rooting Out the Rational, Radical, and Negative 55 Acting Rationally with Fraction-Filled Equations 56 Systematically solving rational equations 56 Solving rational equations with proportions 60 Ridding Yourself of a Radical 61 Squaring both sides of a radical equation 62 Calming two radicals 63 Changing Negative Attitudes about Exponents 65 Flipping negative exponents out of the picture 65 Factoring out negatives to solve equations 66 Fooling Around with Fractional Exponents 68 Combining terms with fractional exponents 69 Factoring fractional exponents 69 Solving equations by working with fractional exponents 70 Chapter 5: Graphing Your Way to the Good Life 73 Coordinating Your Graphing Efforts 74 Identifying the parts of the coordinate plane 74 Plotting from dot to dot 75 Streamlining the Graphing Process with Intercepts and Symmetry 76 Finding x- and y-intercepts 77 Reflecting on a graph’s symmetry 78 Graphing Lines 80 Finding the slope of a line 81 Facing two types of equations for lines 82 Identifying parallel and perpendicular lines 85 Looking at 10 Basic Forms 86 Lines and quadratics 86 Cubics and quartics 87 Radicals and rationals 87 Exponential and logarithmic curves 88 Absolute values and circles 89 Solving Problems with a Graphing Calculator 89 Entering equations into graphing calculators correctly 90 Looking through the graphing window 92 Part 2: Facing Off With Functions 95 Chapter 6: Formulating Function Facts 97 Defining Functions 98 Introducing function notation 98 Evaluating functions 98 Homing In on Domain and Range 99 Determining a function’s domain 99 Describing a function’s range 100 Betting on Even or Odd Functions 102 Recognizing even and odd functions 102 Applying even and odd functions to graphs 103 Facing One-to-One Confrontations 104 Defining one-to-one functions 104 Eliminating one-to-one violators 105 Going to Pieces with Piecewise Functions 106 Doing piecework 107 Applying piecewise functions 108 Composing Yourself and Functions 110 Performing compositions 110 Simplifying the difference quotient 111 Singing Along with Inverse Functions 112 Determining if functions are inverses 112 Solving for the inverse of a function 113 Chapter 7: Sketching and Interpreting Quadratic Functions 115 Interpreting the Standard Form of Quadratics 116 Starting with “a” in the standard form 116 Following up with “b” and “c” 117 Investigating Intercepts in Quadratics 118 Finding the one and only y-intercept 119 Finding the x-intercepts 120 Going to the Extreme: Finding the Vertex 123 Lining Up along the Axis of Symmetry 124 Sketching a Graph from the Available Information 125 Applying Quadratics to the Real World 127 Selling candles 127 Shooting basketballs 128 Launching a water balloon 130 Chapter 8: Staying Ahead of the Curves: Polynomials 133 Taking a Look at the Standard Polynomial Form 134 Exploring Polynomial Intercepts and Turning Points 134 Interpreting relative value and absolute value 135 Counting intercepts and turning points 135 Solving for polynomial intercepts 138 Determining Positive and Negative Intervals 139 Using a sign-line 140 Interpreting the rule 141 Finding the Roots of a Polynomial 143 Factoring for polynomial roots 143 Saving your sanity: The Rational Root Theorem 145 Letting Descartes make a ruling on signs 148 Synthesizing Root Findings 150 Using synthetic division to test for roots 150 Synthetically dividing by a binomial 153 Wringing out the Remainder (Theorem) 154 Chapter 9: Reasoning with Rational Functions 157 Exploring Rational Functions 158 Sizing up domain 158 Introducing intercepts 159 Adding Asymptotes to the Rational Pot 160 Determining the equations of vertical asymptotes 160 Determining the equations of horizontal asymptotes 161 Graphing vertical and horizontal asymptotes 161 Crunching the numbers and graphing oblique asymptotes 163 Accounting for Removable Discontinuities 164 Removal by factoring 164 Evaluating the removal restrictions 165 Showing removable discontinuities on a graph 165 Pushing the Limits of Rational Functions 167 Evaluating limits at discontinuities 168 Going to infinity 170 Catching rational limits at infinity 172 Putting It All Together: Sketching Rational Graphs from Clues 173 Chapter 10: Exposing Exponential and Logarithmic Functions 177 Evaluating Exponential Expressions 178 Exponential Functions: It’s All about the Base, Baby 179 Observing the trends in bases 179 Meeting the most frequently used bases: 10 and e 180 Solving Exponential Equations 182 Making bases match 182 Recognizing and using quadratic patterns 184 Showing an “Interest” in Exponential Functions 186 Applying the compound interest formula 186 Looking at continuous compounding 188 Logging On to Logarithmic Functions 189 Meeting the properties of logarithms 190 Putting your logs to work 191 Solving Logarithmic Equations 193 Setting log equal to log 194 Rewriting log equations as exponentials 195 Graphing Exponential and Logarithmic Functions 196 Expounding on the exponential 196 Not seeing the logs for the trees 198 Part 3: Conquering Conics And Systems Of Equations 203 Chapter 11: Cutting Up Conic Sections 205 Cutting Up a Cone 206 Opening Every Which Way with Parabolas 206 Looking at parabolas with vertices at the origin 207 Observing the general form of parabola equations 210 Sketching the graphs of parabolas 211 Converting parabolic equations to the standard form 214 Going Round and Round in Conic Circles 215 Standardizing the circle 215 Specializing in circles 217 Preparing Your Eyes for Solar Ellipses 218 Raising the standards of an ellipse 218 Sketching an elliptical path 221 Feeling Hyper about Hyperbolas 222 Including the asymptotes 223 Graphing hyperbolas 224 Identifying Conics from Their Equations, Standard or Not 227 Chapter 12: Solving Systems of Linear Equations 229 Looking at the Standard Linear-Systems Form and Its Possible Solutions 230 Graphing Solutions of Linear Systems 230 Pinpointing the intersection 231 Toeing the same line twice 232 Dealing with parallel lines 232 Solving Systems of Two Linear Equations by Using Elimination 233 Getting to the point with elimination 234 Recognizing solutions indicating parallel or coexisting lines 235 Making Substitution the Choice 236 Variable substituting made easy 236 Identifying parallel and coexisting lines 237 Using Cramer’s Rule to Defeat Unwieldy Fractions 238 Setting up the linear system for Cramer 239 Applying Cramer’s Rule to a linear system 240 Tackling Linear Systems with Three Linear Equations 241 Solving three-equation systems with algebra 241 Generalizing multiple solutions for linear equations 243 Upping the Ante with Larger Systems 244 Applying Linear Systems to Our 3-D World 247 Using Systems to Decompose Fractions 248 Chapter 13: Solving Systems of Nonlinear Equations and Inequalities 251 Crossing Parabolas with Lines 252 Determining the point(s) where a line and parabola cross paths 253 Dealing with a solution that’s no solution 254 Intertwining Parabolas and Circles 255 Managing multiple intersections 256 Sorting out the solutions 258 Planning Your Attack on Other Systems of Equations 260 Mixing polynomials and lines 260 Crossing polynomials 261 Navigating exponential intersections 263 Rounding up rational functions 265 Playing Fair with Inequalities 268 Drawing and quartering inequalities 268 Graphing areas with curves and lines 269 Part 4: Shifting Into High Gear With Advanced Concepts 271 Chapter 14: Simplifying Complex Numbers in a Complex World 273 Using Your Imagination to Simplify Powers of i 274 Understanding the Complexity of Complex Numbers 275 Operating on complex numbers 276 Multiplying by the conjugate to perform division 277 Simplifying radicals 279 Solving Quadratic Equations with Complex Solutions 280 Working Polynomials with Complex Solutions 282 Identifying conjugate pairs 283 Interpreting complex zeros 283 Chapter 15: Making Moves with Matrices 287 Describing the Different Types of Matrices 288 Row and column matrices 289 Square matrices 289 Zero matrices 289 Identity matrices 289 Performing Operations on Matrices 290 Adding and subtracting matrices 290 Multiplying matrices by scalars 291 Multiplying two matrices 291 Applying matrices and operations 293 Defining Row Operations 297 Finding Inverse Matrices 298 Determining additive inverses 299 Determining multiplicative inverses 299 Dividing Matrices by Using Inverses 304 Using Matrices to Find Solutions for Systems of Equations 305 Chapter 16: Making a List: Sequences and Series 307 Understanding Sequence Terminology 308 Using sequence notation 308 No-fear factorials in sequences 309 Alternating sequential patterns 309 Looking for sequential patterns 310 Taking Note of Arithmetic and Geometric Sequences 313 Finding common ground: Arithmetic sequences 313 Taking the multiplicative approach: Geometric sequences 315 Recursively Defining Functions 317 Making a Series of Moves 318 Introducing summation notation 318 Summing arithmetically 319 Summing geometrically 320 Applying Sums of Sequences to the Real World 323 Stacking the blocks 323 Negotiating your allowance 323 Bouncing a ball 324 Highlighting Special Formulas 326 Chapter 17: Everything You Wanted to Know about Sets 329 Revealing Set Notation 329 Listing elements with a roster 330 Building sets from scratch 330 Going for all (universal set) or nothing (empty set) 331 Subbing in with subsets 331 Operating on Sets 333 Celebrating the union of two sets 333 Looking both ways for set intersections 334 Feeling complementary about sets 335 Counting the elements in sets 335 Drawing Venn You Feel Like It 336 Applying the Venn diagram 337 Using Venn diagrams with set operations 338 Adding a set to a Venn diagram 339 Focusing on Factorials 342 Making factorial manageable 342 Simplifying factorials 343 How Do I Love Thee? Let Me Count Up the Ways 344 Applying the multiplication principle to sets 344 Arranging permutations of sets 345 Mixing up sets with combinations 348 Branching Out with Tree Diagrams 350 Picturing a tree diagram for a permutation 351 Drawing a tree diagram for a combination 352 Part 5: The Part Of Tens 353 Chapter 18: Ten Multiplication Tricks 355 Squaring Numbers That End in 5 355 Finding the Next Perfect Square 356 Recognizing the Pattern in Multiples of 9 and 11 357 Casting Out 9s 357 Casting Out 9s: The Multiplication Moves 358 Multiplying by 11 359 Multiplying by 5 360 Finding Common Denominators 361 Determining Divisors 362 Multiplying Two-Digit Numbers 362 Chapter 19: Ten Special Types of Numbers 365 Triangular Numbers 365 Square Numbers 366 Hexagonal Numbers 366 Perfect Numbers 367 Amicable Numbers 367 Happy Numbers 368 Abundant Numbers 368 Deficient Numbers 368 Narcissistic Numbers 368 Prime Numbers 369 Index 371

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Book SynopsisTable of ContentsIntroduction 1 Part 1: Getting Off to a Statistically Significant Start 5 Chapter 1: Summarizing Categorical Data: Counts and Percents 7 Chapter 2: Summarizing Quantitative Data: Means, Medians, and More 17 Chapter 3: Organizing Categorical Data: Charts and Graphs 27 Chapter 4: Organizing Quantitative Data: Charts and Graphs 43 Part 2: Probability, Distributions, and the Central Limit Theorem (Are You Having Fun Yet?) 73 Chapter 5: Understanding Probability Basics 75 Chapter 6: Measures of Relative Standing and the Normal Distribution 83 Chapter 7: The Binomial Distribution 105 Chapter 8: The t-Distribution 117 Chapter 9: Demystifying Sampling Distributions and the Central Limit Theorem 123 Part 3: Guesstimating and Hypothesizing with Confidence 137 Chapter 10: Making Sense of Margin of Error 139 Chapter 11: Calculating Confidence Intervals 151 Chapter 12: Deciphering Your Confidence Interval 169 Chapter 13: Testing Hypotheses 177 Chapter 14: Taking the Guesswork Out of p-Values and Type I and II Errors 197 Part 4: Statistical Studies and the Hunt for a Meaningful Relationship 211 Chapter 15: Examining Polls and Surveys 213 Chapter 16: Evaluating Experiments 223 Chapter 17: Looking for Links in Categorical Data: Two-Way Tables 233 Chapter 18: Searching for Links in Quantitative Data: Correlation and Regression 259 Part 5: The Part of Tens 277 Chapter 19: Math Review: Ten Steps to a Better Grade 279 Chapter 20: Top Ten Statistical Formulas 289 Chapter 21: Ten Ways to Spot Common Statistical Mistakes 301 Appendix: Tables for Reference 309 Index 319

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Book SynopsisTable of ContentsAbout the Author vii About the Technical Editors ix Credits xi Acknowledgments xiii Foreword by Nate Vogel xxi Foreword by Sophie Sparkes xxiii Introduction xxv Chapter 1: Introduction and Getting Started with Tableau 1 The Advantages of a Modern Analytics Platform 2 My Personal Tableau Story 3 The Tableau Application Suite 4 Installing Tableau Desktop 5 System Requirements for Tableau Desktop 5 Downloading and Installing Tableau Desktop 6 Registering and Activating Tableau Desktop 6 Data Preparation 6 Crosstab Reports with Wide Tables 7 Preparing Your Data for Analysis 8 Long Tables Suitable for Analysis 8 The Sample Dataset 9 Finding the Dataset 9 Understanding the Data 10 Opening the Excel File Containing the Sample Dataset 11 The Tableau Workspace 13 The Menu Bar 15 The Data Pane 16 Working with Measures and Dimensions 17 Visualizing a First Measure 17 Breaking Down a Measure Based on a Dimension 17 Working with Marks 19 Working with Color 20 Adding More Information to Tooltips 21 Saving, Opening, and Sharing Your Workbooks 21 Saving Workbooks 21 Opening Workbooks 23 Sharing Workbooks with Tableau Reader 23 Chapter 2: Adding Data Sources in Tableau 25 Setting Up a Data Connector 26 Connecting to a File 27 Connecting to a Server 28 Connecting to a Cloud Service 30 Selecting Data Tables 31 Adding a Table to a Data Model 31 Joins 31 Unions 34 Specific Unions (Manual) 34 Wildcard Unions (Automatic) 35 Data Extracts and Live Connections 36 Live Connections 36 Untethered with a Data Extract 37 Data Protection and Data Governance 38 Editing the Model’s Metadata 38 Data Types 40 Changing a Field’s Data Type 41 Adding Hierarchies, Calculated Fields, and Table Calculations 41 Data Collection 42 Data Collection with IFTTT and Google Sheets 42 Website Analysis with Google Analytics 43 Checklist for Increasing Performance 46 General Advice for Performance Optimization 46 Performance Optimization with Files and Cloud Services 47 Performance Optimization with Database Servers 48 Chapter 3: Creating Data Visualizations 49 Chart Types 50 Ready, Set, Show Me 52 How Show Me Works 52 Scatter Plots 52 Bar Charts, Legends, Filters, and Hierarchies 53 Bar Charts 54 Hierarchies 54 Filters 55 Line Charts 57 Straight Lines 57 Adjusting the Time Dimension 58 Step Charts 59 Jump Lines 59 Continuous Date Fields 61 Highlight Tables 64 Step 1: Cross Tables 64 Step 2: Add Color 64 Step 3: Change the Mark Type 66 Heatmaps 67 Step 1: Build the Table 67 Step 2: Choose an Interesting Color Palette 68 Step 3: Change the Size of Marks 69 Bullet Charts 69 Step 1: Side-By-Side Bars 71 Step 2: Overlay the Measures 71 Cumulative Sums with Waterfall Charts 73 Step 1: Sorted Bar Chart 73 Step 2: Cumulative Sum and Gantt bars 74 Step 3: Calculate the Step Size 75 Reflection: The Anatomy of a Tableau Visualization 77 Chapter 4: Aggregate Functions, Calculated Fields, and Parameters 81 Aggregate Functions 82 Calculated Fields 84 Aggregations in Calculated Fields 86 Text Operators 88 Splits 88 Shortening Character Strings 89 Converting Between Uppercase and Lowercase 90 Replacing Substrings 90 Date Fields 90 Date Parts 90 Traditional Gregorian and ISO 8601 Calendars 91 Date Calculations 92 Parsing Date Parts 92 Date Format Conversions 93 Logical Functions in Calculated Fields 94 Case Discrimination 94 Case Discrimination with IF-THEN-ELSE Logic 95 Case Discrimination with the IIF Function 96 Working with NULL Values 97 Parameters 97 Creating a Parameter and Displaying the Control Element 97 Parameters in Calculated Fields 98 Searching Text Fields 100 Chapter 5: Table Calculations and Level of Detail Calculations 105 Different Types of Calculations 106 Order of Processing Steps 107 Quick Table Calculations 107 Setting Up a Quick Table Calculation 107 Duplicate as Crosstab 110 Editing Table Calculations 110 Customized Table Calculations 113 Bump Charts 113 Dual Axis Charts 116 Adjustable Moving Average 118 Level of Detail Expressions 123 Keywords and Syntax 123 Cohort Analysis 124 Regional Averages 125 Higher-Level Regions 127 Chapter 6: Maps 131 Symbol Maps 132 Filled Maps 134 Density Maps 134 Map Layers 136 Maps with Pie Charts 137 Creating a Pie Chart Map 138 Adding a Filter 138 Dual Axis Map 139 Viz in Tooltip 140 Step 1: Create the Second Chart 141 Step 2: Embedding the Chart in Tooltips 142 Reflection: The Anatomy of a Tableau Map 143 Alternative Map Services 144 Mapbox Maps 145 Mapbox Account and Token 145 Mapbox in Tableau 146 Using the Background Map 146 Spatial Data 147 Undersea Communication Cables 148 Open Data 152 Chapter 7: Advanced Analytics: Trends, Forecasts, Clusters, and other Statistical Tools 155 Overview of the Tableau Analytics Pane 156 Constant, Average, and Reference Lines 157 Trend Lines 157 Adding Trend Lines 158 Trend Line Options 160 Line and Trend Model Description 161 Forecasts 162 Adding a Forecast Line to the View 162 Forecast Settings 162 Model Description 164 Cluster Analysis 166 Clustering in Tableau 166 Saving and Working with Clustering Results 167 Python, R, and MATLAB Integration 168 Getting Started with Python and TabPy 169 Connecting Tableau with TabPy 170 Python Scripts in Calculated Fields 172 Trellis Chart with Python Script 173 R Integration 174 Security 175 Example: Local Regression with R 175 MATLAB Integration 178 Chapter 8: Interactive Dashboards 181 Preliminary Considerations 182 Creating a New Dashboard 183 The Dashboard Pane 184 Placing Charts on the Dashboard 185 Dashboard Titles 187 Navigation Buttons 188 Dashboard Actions 191 Filter Actions 191 Adding and Editing Filter and Highlight Actions 193 Adding Web Content via URL Actions 195 Email Notifications via URL Actions 198 Dashboard Starters: Templates for Cloud Data 199 Dashboard Best Practices and Inspiration 201 Design Tips for Creating a Dashboard 201 Tableau Public: A Gallery of Inspiration 202 Chapter 9: Sharing Insights with Colleagues and the World 205 Preliminary Considerations 206 Tableau Online and Tableau Server 207 Publishing 207 Ask Data 211 Tableau Mobile 212 Tableau Public 213 Publishing to Tableau Public 214 Your Tableau Public Profile 216 Web Embedding 216 Chapter 10: Data Preparation with Tableau Prep 221 Connecting to Data 222 Wildcard Unions 226 Additional Connections 228 Inspecting the Data 229 Removing Unneeded Fields 230 Data Cleaning and Formatting 231 Cleaning Steps and the Profile Pane 231 Calculated Fields 233 Built-in Cleaning Features 235 Renaming Cleaning Steps 235 Unions 237 Joins 238 Splits 239 Grouping 240 Joining 240 Running the Flow and Outputting the Data 242 Saving Flows 244 Index 245

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Book SynopsisHelp your students to think critically and creatively through team-based problem solving instead of focusing on testing and outcomes. Professionals throughout the education system are recognizing that standardized testing is holding students back. Schools tend to view children as outcomes rather than as individuals who require guidance on thinking critically and creatively. Awesome Math focuses on team-based problem solving to teach discrete mathematics, a subject essential for success in the STEM careers of the future. Built on the increasingly popular growth mindset, this timely book emphasizes a problem-solving approach for developing the skills necessary to think critically, creatively, and collaboratively. In its current form, math education is a series of exercises: straightforward problems with easily-obtained answers. Problem solving, however, involves multiple creative approaches to solving meaningful and interesting problems. The authors, co-founders of the multi-layered eTable of ContentsAcknowledgments xi About the Authors xiii Introduction xvii I. Why Problem Solving? Chapter 1: Rewards for Problem-Based Approach: Range, Rigor, and Resilience 5 Range Ignites Curiosity 5 Rigor Taps Critical Thinking 9 Resilience is Born Through Creativity 10 Chapter 2: Maximize Learning: Relevance, Authenticity, and Usefulness 13 Student Relevance 13 Mathematical Relevance 14 Mathematical Relevance: The Math Circle Example 16 Curriculum Relevance 18 Authenticity: The Cargo Cult Science Trap 21 Authenticity in Learning 22 Usefulness 25 Chapter 3: Creating a Math Learning Environment 27 Know Yourself: Ego and Grace 27 Know Your Students 30 Know Your Approach 35 Chapter 4: What is the Telos? 47 Autonomy to Solve Your Problems 47 Mastery Through Inquiry 48 Purpose with Competitions 50 Quadrants of Success 52 Chapter 5: Gains and Pains with a Problem-Based Curriculum 57 Teachers 58 Students 61 Parents 67 II. Teaching Problem Solving Chapter 6: Five Steps to Problem-Based Learning 75 Start with Meaningful Problems 75 Utilize Teacher Resources 79 Provide an Active Learning Environment 91 Understand the Value of Mistakes 97 Recognize That Everyone is Good at Math 99 Chapter 7: The Three Cs: Competitions, Collaboration, Community 103 Competitions 103 Collaboration 107 Community 117 Aspire to Inspire: Stories from Awesome Educators 121 Chapter 8: Mini-Units 147 Relate/Reflect/Revise Questions 147 Roman Numeral Problems 148 Cryptarithmetic 151 Squaring Numbers: Mental Mathematics 155 The Number of Elements of a Finite Set 157 Magic Squares 159 Toothpicks Math 163 Pick’s Theorem 165 Equilateral versus Equiangular 168 Math and Chess 170 Area and Volume of a Sphere 172 III. Full Units Chapter 9: Angles and Triangles 177 Learning Objectives 177 Definitions 177 Angles and Parallel Lines 177 Summary 180 Chapter 10: Consecutive Numbers 185 Learning Objectives 185 Definitions 185 Chapter 11: Factorials! 191 Learning Objectives 191 Definitions 191 Chapter 12: Triangular Numbers 199 Learning Objectives 199 Definitions 199 Chapter 13: Polygonal Numbers 205 Learning Objectives 205 Definitions 205 Chapter 14: Pythagorean Theorem Revisited 213 Learning Objectives 213 Definitions 213 Pythagorean Theorem 214 Rectangular Boxes 214 Euler Bricks 216 Assessment Problems 219 Chapter 15: Sequences 221 Learning Objectives 221 Definitions 221 Introduce a Geometric Progression 222 Chapter 16: Pigeonhole Principle 227 Learning Objectives 227 Definitions 227 Chapter 17: Viviani’s Theorems 235 Learning Objectives 235 Definition 235 Chapter 18: Dissection Time 239 Learning Objectives 239 Definitions 239 Chapter 19: Pascal’s Triangle 245 Learning Objective 245 Summary 249 Chapter 20: Nice Numbers 255 Learning Objectives 255 Definitions 255 Index 259

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Book SynopsisTable of ContentsIntroduction 1 Chapter 1: Setting the Scene for Actions in Algebra 5 Chapter 2: Examining Powers and Roots 17 Chapter 3: Ordering and Distributing: The Business of Algebra 25 Chapter 4: Factoring in the First and Second Degrees 33 Chapter 5: Broadening the Factoring Horizon 45 Chapter 6: Solving Linear Equations 57 Chapter 7: Tackling Second-Degree Quadratic Equations 71 Chapter 8: Expanding the Equation Horizon 87 Chapter 9: Reconciling Inequalities 103 Chapter 10: Absolute-Value Equations and Inequalities 115 Chapter 11: Making Algebra Tell a Story 121 Chapter 12: Putting Geometry into Story Problems 133 Chapter 13: Grappling with Graphing 143 Chapter 14: Ten Warning Signs of Algebraic Pitfalls 157 Index 161

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Book SynopsisProvides clear guidance on leveraging computational techniques to answer social science questions In disciplines such as political science, sociology, psychology, and media studies, the use of computational analysis is rapidly increasing. Statistical modeling, machine learning, and other computational techniques are revolutionizing the way electoral results are predicted, social sentiment is measured, consumer interest is evaluated, and much more. Computational Analysis of Communication teaches social science students and practitioners how computational methods can be used in a broad range of applications, providing discipline-relevant examples, clear explanations, and practical guidance. Assuming little or no background in data science or computer linguistics, this accessible textbook teaches readers how to use state-of-the art computational methods to perform data-driven analyses of social science issues. A cross-disciplinary team of authorswith expertise in both the social sciencTable of ContentsPreface xi Acknowledgments xiii 1 Introduction 1 1.1 The Role of Computational Analysis in the Social Sciences 1 1.2 Why Python and/or R? 3 1.3 How to Use This Book 4 1.4 Installing R and Python 5 1.4.1 Installing R and RStudio 7 1.4.2 Installing Python and Jupyter Notebook 9 1.5 Installing Third-Party Packages 12 2 Getting Started: Fun with Data and Visualizations 13 2.1 Fun With Tweets 14 2.2 Fun With Textual Data 15 2.3 Fun With Visualizing Geographic Information 17 2.4 Fun With Networks 19 3 Programming Concepts for Data Analysis 23 3.1 About Objects and Data Types 24 3.1.1 Storing Single Values: Integers, Floating-Point Numbers, Booleans 25 3.1.2 Storing Text 26 3.1.3 Combining Multiple Values: Lists, Vectors, And Friends 28 3.1.4 Dictionaries 32 3.1.5 From One to More Dimensions: Matrices and n-Dimensional Arrays 33 3.1.6 Making Life Easier: Data Frames 34 3.2 Simple Control Structures: Loops and Conditions 35 3.2.1 Loops 36 3.2.2 Conditional Statements 37 3.3 Functions and Methods 39 4 How to Write Code 43 4.1 Re-using Code: How Not to Re-Invent the Wheel 43 4.2 Understanding Errors and Getting Help 46 4.2.1 Error Messages 46 4.2.2 Debugging Strategies 48 4.3 Best Practice: Beautiful Code, GitHub, and Notebooks 49 5 From File to Data Frame and Back 55 5.1 Why and When Do We Use Data Frames? 56 5.2 Reading and Saving Data 57 5.2.1 The Role of Files 57 5.2.2 Encodings and Dialects 59 5.2.3 File Handling Beyond Data Frames 61 5.3 Data from Online Sources 62 6 Data Wrangling 65 6.1 Filtering, Selecting, and Renaming 66 6.2 Calculating Values 67 6.3 Grouping and Aggregating 69 6.3.1 Combining Multiple Operations 70 6.3.2 Adding Summary Values 71 6.4 Merging Data 72 6.4.1 Equal Units of Analysis 72 6.4.2 Inner and Outer Joins 75 6.4.3 Nested Data 76 6.5 Reshaping Data: Wide To Long And Long To Wide 78 6.6 Restructuring Messy Data 79 7 Exploratory Data Analysis 83 7.1 Simple Exploratory Data Analysis 84 7.2 Visualizing Data 87 7.2.1 Plotting Frequencies and Distributions 88 7.2.2 Plotting Relationships 92 7.2.3 Plotting Geospatial Data 98 7.2.4 Other Possibilities 99 7.3 Clustering and Dimensionality Reduction 100 7.3.1 k-means Clustering 101 7.3.2 Hierarchical Clustering 102 7.3.3 Principal Component Analysis and Singular Value Decomposition 106 8 Statistical Modeling and Supervised Machine Learning 113 8.1 Statistical Modeling and Prediction 115 8.2 Concepts and Principles 117 8.3 Classical Machine Learning: From Naïve Bayes to Neural Networks 122 8.3.1 Naïve Bayes 122 8.3.2 Logistic Regression 124 8.3.3 Support Vector Machines 125 8.3.4 Decision Trees and Random Forests 127 8.3.5 Neural Networks 129 8.4 Deep Learning 130 8.4.1 Convolutional Neural Networks 131 8.5 Validation and Best Practices 133 8.5.1 Finding a Balance Between Precision and Recall 133 8.5.2 Train, Validate, Test 137 8.5.3 Cross-validation and Grid Search 138 9 Processing Text 141 9.1 Text as a String of Characters 142 9.1.1 Methods for Dealing With Text 144 9.2 Regular Expressions 145 9.2.1 Regular Expression Syntax 146 9.2.2 Example Patterns 147 9.3 Using Regular Expressions in Python and R 150 9.3.1 Splitting and Joining Strings, and Extracting Multiple Matches 151 10 Text as Data 155 10.1 The Bag of Words and the Term-Document Matrix 156 10.1.1 Tokenization 157 10.1.2 The DTM as a Sparse Matrix 159 10.1.3 The DTM as a “Bag of Words” 162 10.1.4 The (Unavoidable) Word Cloud 163 10.2 Weighting and Selecting Documents and Terms 164 10.2.1 Removing stop words 165 10.2.2 Removing Punctuation and Noise 167 10.2.3 Trimming a DTM 170 10.2.4 Weighting a DTM 171 10.3 Advanced Representation of Text 172 10.3.1 n-grams 173 10.2.3 Collocations 174 10.3.3 Word Embeddings 176 10.3.4 Linguistic Preprocessing 177 10.4 Which Preprocessing to Use? 182 11 Automatic Analysis of Text 184 11.1 Deciding on the Right Method 185 11.2 Obtaining a Review Dataset 187 11.3 Dictionary Approaches to Text Analysis 189 11.4 Supervised Text Analysis: Automatic Classification and Sentiment Analysis 191 11.4.1 Putting Together a Workflow 191 11.4.2 Finding the Best Classifier 194 11.4.3 Using the Model 198 11.4.4 Deep Learning 199 11.5 Unsupervised Text Analysis: Topic Modeling 203 11.5.1 Latent Dirichlet Allocation (LDA) 203 11.5.2 Fitting an LDA Model 206 11.5.3 Analyzing Topic Model Results 207 11.5.4 Validating and Inspecting Topic Models 208 11.5.5 Beyond LDA 209 12 Scraping Online Data 212 12.1 Using Web APIs: From Open Resources to Twitter 213 12.2 Retrieving and Parsing Web Pages 219 12.2.1 Retrieving and Parsing an HTML Page 219 12.2.2 Crawling Websites 223 12.2.3 Dynamic Web Pages 225 12.3 Authentication, Cookies, and Sessions 228 12.3.1 Authentication and APIs 228 12.3.2 Authentication and Webpages 229 12.4 Ethical, Legal, and Practical Considerations 230 13 Network Data 233 13.1 Representing and Visualizing Networks 234 13.2 Social Network Analysis 241 13.2.1 Paths and Reachability 242 13.2.2 Centrality Measures 246 13.2.3 Clustering and Community Detection 248 14 Multimedia Data 258 14.1 Beyond Text Analysis: Images, Audio and Video 259 14.2 Using Existing Libraries and APIs 261 14.3 Storing, Representing, and Converting Images 263 14.4 Image Classification 270 14.4.1 Basic Classification with Shallow Algorithms 272 14.4.2 Deep Learning for Image Analysis 273 14.4.3 Re-using an Open Source CNN 279 15 Scaling Up and Distributing 283 15.1 Storing Data in SQL and noSQL Databases 283 15.1.1 When to Use a Database 283 15.1.2 Choosing the Right Database 285 15.1.3 A Brief Example Using SQLite 286 15.2 Using Cloud Computing 286 15.3 Publishing Your Source 290 15.4 Distributing Your Software as Container 291 16 Where to Go Next 293 16.1 How Far Have We Come? 293 16.2 Where To Go Next? 294 16.3 Open, Transparent, and Ethical Computational Science 295 Bibliography 297 Index 303

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Book SynopsisPRICING INSURANCE RISK A comprehensive framework for measuring, valuing, and managing risk Pricing Insurance Risk: Theory and Practice delivers an accessible and authoritative account of how to determine the premium for a portfolio of non-hedgeable insurance risks and how to allocate it fairly to each portfolio component. The authors synthesize hundreds of academic research papers, bringing to light little-appreciated answers to fundamental questions about the relationships between insurance risk, capital, and premium. They lean on their industry experience throughout to connect the theory to real-world practice, such as assessing the performance of business units, evaluating risk transfer options, and optimizing portfolio mix. Readers will discover: Definitions, classifications, and specifications of riskAn in-depth treatment of classical risk measures and premium calculation principlesProperties of risk measures and their visualizationA logical framework for spectral and coherentTable of ContentsPreface xii 1 Introduction 1 1.1 Our Subject and Why It Matters 1 1.2 Players, Roles, and Risk Measures 2 1.3 Book Contents and Structure 4 1.4 What’s in It for the Practitioner? 7 1.5 Where to Start 9 2 The Insurance Market and Our Case Studies 13 2.1 The Insurance Market 13 2.2 Ins Co.: A One-Period Insurer 15 2.3 Model vs. Reality 16 2.4 Examples and Case Studies 17 2.5 Learning Objectives 25 Part I Risk 27 3 Risk and Risk Measures 29 3.1 Risk in Everyday Life 29 3.2 Defining Risk 30 3.3 Taxonomies of Risk 31 3.4 Representing Risk Outcomes 36 3.5 The Lee Diagram and Expected Losses 40 3.6 Risk Measures 54 3.7 Learning Objectives 60 4 Measuring Risk with Quantiles, VaR, and TVaR 63 4.1 Quantiles 63 4.2 Value at Risk 70 4.3 Tail VaR and Related Risk Measures 85 4.4 Differentiating Quantiles, VaR, and TVaR 102 4.5 Learning Objectives 102 5 Properties of Risk Measures and Advanced Topics 105 5.1 Probability Scenarios 105 5.2 Mathematical Properties of Risk Measures 110 5.3 Risk Preferences 124 5.4 The Representation Theorem for Coherent Risk Measures 130 5.5 Delbaen’s Differentiation Theorem 137 5.6 Learning Objectives 141 5.A Lloyd’s Realistic Disaster Scenarios 142 5.B Convergence Assumptions for Random Variables 143 6 Risk Measures in Practice 147 6.1 Selecting a Risk Measure Using the Characterization Method 147 6.2 Risk Measures and Risk Margins 148 6.3 Assessing Tail Risk in a Univariate Distribution 149 6.4 The Intended Purpose: Applications of Risk Measures 150 6.5 Compendium of Risk Measures 153 6.6 Learning Objectives 156 7 Guide to the Practice Chapters 157 Part II Portfolio Pricing 161 8 Classical Portfolio Pricing Theory 163 8.1 Insurance Demand, Supply, and Contracts 163 8.2 Insurer Risk Capital 168 8.3 Accounting Valuation Standards 178 8.4 Actuarial Premium Calculation Principles and Classical Risk Theory 182 8.5 Investment Income in Pricing 186 8.6 Financial Valuation and Perfect Market Models 189 8.7 The Discounted Cash Flow Model 192 8.8 Insurance Option Pricing Models 200 8.9 Insurance Market Imperfections 210 8.10 Learning Objectives 213 8.A Short- and Long-Duration Contracts 215 8.B The Equivalence Principle 216 9 Classical Portfolio Pricing Practice 217 9.1 Stand-Alone Classical PCPs 217 9.2 Portfolio CCoC Pricing 223 9.3 Applications of Classical Risk Theory 224 9.4 Option Pricing Examples 227 9.5 Learning Objectives 231 10 Modern Portfolio Pricing Theory 233 10.1 Classical vs. Modern Pricing and Layer Pricing 233 10.2 Pricing with Varying Assets 235 10.3 Pricing by Layer and the Layer Premium Density 238 10.4 The Layer Premium Density as a Distortion Function 239 10.5 From Distortion Functions to the Insurance Market 245 10.6 Concave Distortion Functions 252 10.7 Spectral Risk Measures 255 10.8 Properties of an SRM and Its Associated Distortion Function 259 10.9 Six Representations of Spectral Risk Measures 261 10.10 Simulation Interpretation of Distortion Functions 263 10.11 Learning Objectives 264 10.A Technical Details 265 11 Modern Portfolio Pricing Practice 271 11.1 Applying SRMs to Discrete Random Variables 271 11.2 Building-Block Distortions and SRMs 275 11.3 Parametric Families of Distortions 280 11.4 SRM Pricing 285 11.5 Selecting a Distortion 292 11.6 Fitting Distortions to Cat Bond Data 298 11.7 Resolving an Apparent Pricing Paradox 304 11.8 Learning Objectives 306 Part III Price Allocation 307 12 Classical Price Allocation Theory 309 12.1 The Allocation of Portfolio Constant CoC Pricing 309 12.2 Allocation of Non-Additive Functionals 312 12.3 Loss Payments in Default 324 12.4 The Historical Development of Insurance Pricing Models 326 12.5 Learning Objectives 337 13 Classical Price Allocation Practice 339 13.1 Allocated CCoC Pricing 339 13.2 Allocation of Classical PCP Pricing 347 13.3 Learning Objectives 348 14 Modern Price Allocation Theory 349 14.1 The Natural Allocation of a Coherent Risk Measure 349 14.2 Computing the Natural Allocations 365 14.3 A Closer Look at Unit Funding 369 14.4 An Axiomatic Approach to Allocation 385 14.5 Axiomatic Characterizations of Allocations 392 14.6 Learning Objectives 394 15 Modern Price Allocation Practice 397 15.1 Applying the Natural Allocations to Discrete Random Variables 397 15.2 Unit Funding Analysis 404 15.3 Bodoff’s Percentile Layer of Capital Method 413 15.4 Case Study Exhibits 421 15.5 Learning Objectives 439 Part IV Advanced Topics 441 16 Asset Risk 443 16.1 Background 443 16.2 Adding Asset Risk to Ins Co. 444 16.3 Learning Objectives 447 17 Reserves 449 17.1 Time Periods and Notation 449 17.2 Liability for Ultimate Losses 450 17.3 The Solvency II Risk Margin 461 17.4 Learning Objectives 468 18 Going Concern Franchise Value 469 18.1 Optimal Dividends 469 18.2 The Firm Life Annuity 472 18.3 Learning Objectives 476 19 Reinsurance Optimization 477 19.1 Background 477 19.2 Evaluating Ceded Reinsurance 477 19.3 Learning Objectives 481 20 Portfolio Optimization 483 20.1 Strategic Framework 483 20.2 Market Regulation 484 20.3 Dynamic Capital Allocation and Marginal Cost 485 20.4 Marginal Cost and Marginal Revenue 487 20.5 Performance Management and Regulatory Rigidities 488 20.6 Practical Implications 490 20.7 Learning Objectives 491 A Background Material 493 A.1 Interest Rate, Discount Rate, and Discount Factor 493 A.2 Actuarial vs. Accounting Sign Conventions 493 A.3 Probability Theory 494 A.4 Additional Mathematical Terminology 500 B Notation 503 References 507 Index 523

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