|This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory’s historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts, states all necessary results, and provides relevant proofs. The second part opens with definitions of probabilistic metric spaces and proceeds to examinations of special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. Throughout, the authors focus on developing aspects that differ from the theory of ordinary metric spaces, rather than simply transferring known metric space results to a more general setting.
|The utility of artificial neural network models lies in the fact that they can be used to infer functions from observations—making them especially useful in applications where the complexity of data or tasks makes the design of such functions by hand impractical. Exploring Neural Networks with C# presents the important properties of neural networks—while keeping the complex mathematics to a minimum. Explaining how to build and use neural networks, it presents complicated information about neural networks structure, functioning, and learning in a manner that is easy to understand. Taking a ‘learn by doing’ approach, the book is filled with illustrations to guide you through the mystery of neural networks. Examples of experiments are provided in the text to encourage individual research. Online access to C# programs is also provided to help you discover the properties of neural networks. Following the procedures and using the programs included with the book will allow you to learn how to work with neural networks and evaluate your progress. You can download the programs as both executable applications and C# source code from http://home.agh.edu.pl/~tad//index.php?page=programy&lang=en|
|This book introduces students to probability, statistics, and stochastic processes. It can be used by both students and practitioners in engineering, various sciences, finance, and other related fields. It provides a clear and intuitive approach to these topics while maintaining mathematical accuracy The book covers:
• Basic concepts such as random experiments, probability axioms, conditional probability, and counting methods
• Single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities
• Limit theorems and convergence
• Introduction to Bayesian and classical statistics
• Random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion
• Simulation using MATLAB and R (online chapters)
The book contains a large number of solved exercises. The dependency between different sections of this book has been kept to a minimum in order to provide maximum flexibility to instructors and to make the book easy to read for students. Examples of applications—such as engineering, finance, everyday life, etc.—are included to aid in motivating the subject. The digital version of the book, as well as additional materials such as videos, is available at http://www.probabilitycourse.com.