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Dieses Lehrbuch umfaßt einen Kanon von Themen, der an vielen Universitäten unter dem Titel "Diskrete Strukturen" fester Bestandteil des Informatik-Grundstudiums geworden ist. Bei der Darstellung wird neben der mathematischen Exaktheit besonderer Wert darauf gelegt, auch das intuitive Verständnis zu fördern, um so das Verstehen und Einordnen des Stoffs zu erleichtern. Unterstützt wird dies durch zahlreiche Beispiele und Aufgaben, vorwiegend aus dem Bereich der Informatik. Themen: Kombinatorik, Graphentheorie, Algorithmische Grundprinzipien, Rekursionsgleichungen, Algebra.

Computer science—Mathematics. --- Mathematical statistics. --- Discrete Mathematics in Computer Science. --- Probability and Statistics in Computer Science. --- Math Applications in Computer Science.

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In diesem Buch werden ausführlich reale Anwendungen der Statistik in Wirtschaft und Forschung erörtert und mit einer exakten Herleitung der vorgestellten Verfahren verbunden. Damit baut das Buch eine breite Brücke zwischen Theorie und Praxis der Statistik. Die zahlreichen Anwendungsbeispiele werden mit dem kostenlosen Softwarepaket R und der zugehörigen grafischen Oberfläche R-Commander berechnet. Zusätzlich werden Projekte an der Schnittstelle zwischen Schul- und Hochschulunterricht vorgestellt, die mit Schülern oder Studenten durchgeführt werden können. Das Buch spricht insbesondere Lehrer und Studierende des Lehramts an Gymnasien an. Für die Lektüre werden lediglich Stochastik-Kenntnisse vorausgesetzt, wie sie in entsprechenden Einführungsvorlesungen vermittelt werden. Kenntnisse des Programmpakets R sind nicht notwendig, da alle durchgeführten Schritte ausführlich erklärt werden und zusätzlich in einem separaten Teil des Buchs eine Einführung in R zu finden ist.

Statistics. --- Computer science --- Mathematical statistics. --- Mathematics. --- Probabilities. --- Statistical Theory and Methods. --- Probability and Statistics in Computer Science. --- Applications of Mathematics. --- Probability Theory.

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Monte Carlo Methods are among the most used, and useful, computational tools available today. They provide efficient and practical algorithms to solve a wide range of scientific and engineering problems in dozens of areas many of which are covered in this text. These include simulation, optimization, finance, statistical mechanics, birth and death processes, Bayesian inference, quadrature, gambling systems and more. This text is for students of engineering, science, economics and mathematics who want to learn about Monte Carlo methods but have only a passing acquaintance with probability theory. The probability needed to understand the material is developed within the text itself in a direct manner using Monte Carlo experiments for reinforcement. There is a prerequisite of at least one year of calculus and a semester of matrix algebra. Each new idea is carefully motivated by a realistic problem, thus leading to insights into probability theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. All examples in the text are coded in Python as a representative language; the logic is sufficiently clear so as to be easily translated into any other language. Further, Python scripts for each worked example are freely accessible for each chapter. Along the way, most of the basic theory of probability is developed in order to illuminate the solutions to the questions posed. One of the strongest features of the book is the wealth of completely solved example problems. These provide the reader with a sourcebook to follow towards the solution of their own computational problems. Each chapter ends with a large collection of homework problems illustrating and directing the material. This book is suitable as a textbook for students of engineering, finance, and the sciences as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability as well as for a more specialized course in Monte Carlo Methods. Topics include probability distributions, probability calculations, sampling, counting combinatorial objects, Markov chains, random walks, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, random number generation, Bayesian Inference, Gibbs Sampling and Monte Carlo integration.

Probabilities. --- Computer science --- Mathematical statistics. --- Algorithms. --- Game theory. --- Computer simulation. --- Mathematical optimization. --- Probability Theory. --- Probability and Statistics in Computer Science. --- Game Theory. --- Computer Modelling. --- Optimization. --- Mathematics.

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This first comprehensive book on models behind Generative AI has been thoroughly revised to cover all major classes of deep generative models: mixture models, Probabilistic Circuits, Autoregressive Models, Flow-based Models, Latent Variable Models, GANs, Hybrid Models, Score-based Generative Models, Energy-based Models, and Large Language Models. In addition, Generative AI Systems are discussed, demonstrating how deep generative models can be used for neural compression. All chapters are accompanied by code snippets that help to better understand the modeling frameworks presented. Deep Generative Modeling is designed to appeal to curious students, engineers, and researchers with a modest mathematical background in undergraduate calculus, linear algebra, probability theory, and the basics of machine learning, deep learning, and programming in Python and PyTorch (or other deep learning libraries). It should appeal to students and researchers from a variety of backgrounds, including computer science, engineering, data science, physics, and bioinformatics who wish to get familiar with deep generative modeling. In order to engage with a reader, the book introduces fundamental concepts with specific examples and code snippets. The full code accompanying the book is available on the author's GitHub site: github.com/jmtomczak/intro_dgm The ultimate aim of the book is to outline the most important techniques in deep generative modeling and, eventually, enable readers to formulate new models and implement them.

Artificial intelligence. --- Machine learning. --- Computer science --- Mathematical statistics. --- Computer simulation. --- Artificial Intelligence. --- Machine Learning. --- Probability and Statistics in Computer Science. --- Computer Modelling. --- Mathematics.

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Applied Probability presents a unique blend of theory and applications, with special emphasis on mathematical modeling, computational techniques, and examples from the biological sciences. Chapter 1 reviews elementary probability and provides a brief survey of relevant results from measure theory. Chapter 2 is an extended essay on calculating expectations. Chapter 3 deals with probabilistic applications of convexity, inequalities, and optimization theory. Chapters 4 and 5 touch on combinatorics and combinatorial optimization. Chapters 6 through 11 present core material on stochastic processes. If supplemented with appropriate sections from Chapters 1 and 2, there is sufficient material for a traditional semester-long course in stochastic processes covering the basics of Poisson processes, Markov chains, branching processes, martingales, and diffusion processes. This third edition includes new topics and many worked exercises. The new chapter on entropy stresses Shannon entropy and its mathematical applications. New sections in existing chapters explain the Chinese restaurant problem, the infinite alleles model, saddlepoint approximations, and recurrence relations. The extensive list of new problems pursues topics such as random graph theory omitted in the previous editions. Computational probability receives even greater emphasis than earlier. Some of the solved problems are coding exercises, and Julia code is provided. Mathematical scientists from a variety of backgrounds will find Applied Probability appealing as a reference. This updated edition can serve as a textbook for graduate students in applied mathematics, biostatistics, computational biology, computer science, physics, and statistics. Readers should have a working knowledge of multivariate calculus, linear algebra, ordinary differential equations, and elementary probability theory.

Statistics. --- Probabilities. --- Computer science --- Mathematical statistics. --- Mathematics --- Statistical Theory and Methods. --- Probability Theory. --- Probability and Statistics in Computer Science. --- Computational Mathematics and Numerical Analysis. --- Mathematics. --- Data processing.