- Table View
- List View
Probabilistic Approaches for Geotechnical Site Characterization and Slope Stability Analysis
by Dianqing Li Yu Wang Zijun CaoThis is the first book to revisit geotechnical site characterization from a probabilistic point of view and provide rational tools to probabilistically characterize geotechnical properties and underground stratigraphy using limited information obtained from a specific site. This book not only provides new probabilistic approaches for geotechnical site characterization and slope stability analysis, but also tackles the difficulties in practical implementation of these approaches. In addition, this book also develops efficient Monte Carlo simulation approaches for slope stability analysis and implements these approaches in a commonly available spreadsheet environment. These approaches and the software package are readily available to geotechnical practitioners and alleviate them from reliability computational algorithms. The readers will find useful information for a non-specialist to determine project-specific statistics of geotechnical properties and to perform probabilistic analysis of slope stability.
Probabilistic Cellular Automata: Theory, Applications And Future Perspectives (Emergence, Complexity And Computation Ser. #27)
by Pierre-Yves Louis Francesca R. NardiThis book explores Probabilistic Cellular Automata (PCA) from the perspectives of statistical mechanics, probability theory, computational biology and computer science. PCA are extensions of the well-known Cellular Automata models of complex systems, characterized by random updating rules. Thanks to their probabilistic component, PCA offer flexible computing tools for complex numerical constructions, and realistic simulation tools for phenomena driven by interactions among a large number of neighboring structures. PCA are currently being used in various fields, ranging from pure probability to the social sciences and including a wealth of scientific and technological applications. This situation has produced a highly diversified pool of theoreticians, developers and practitioners whose interaction is highly desirable but can be hampered by differences in jargon and focus. This book – just as the workshop on which it is based – is an attempt to overcome these difference and foster interest among newcomers and interaction between practitioners from different fields. It is not intended as a treatise, but rather as a gentle introduction to the role and relevance of PCA technology, illustrated with a number of applications in probability, statistical mechanics, computer science, the natural sciences and dynamical systems. As such, it will be of interest to students and non-specialists looking to enter the field and to explore its challenges and open issues.
Probabilistic Data Structures for Blockchain-Based Internet of Things Applications
by Neeraj Kumar Arzoo MiglaniThis book covers theory and practical knowledge of Probabilistic data structures (PDS) and Blockchain (BC) concepts. It introduces the applicability of PDS in BC to technology practitioners and explains each PDS through code snippets and illustrative examples. Further, it provides references for the applications of PDS to BC along with implementation codes in python language for various PDS so that the readers can gain confidence using hands on experience. Organized into five sections, the book covers IoT technology, fundamental concepts of BC, PDS and algorithms used to estimate membership query, cardinality, similarity and frequency, usage of PDS in BC based IoT and so forth.
Probabilistic Finite Element Model Updating Using Bayesian Statistics: Applications to Aeronautical and Mechanical Engineering
by Tshilidzi Marwala Ilyes Boulkaibet Sondipon AdhikariProbabilistic Finite Element Model Updating Using Bayesian Statistics: Applications to Aeronautical and Mechanical Engineering Tshilidzi Marwala and Ilyes Boulkaibet, University of Johannesburg, South Africa Sondipon Adhikari, Swansea University, UK Covers the probabilistic finite element model based on Bayesian statistics with applications to aeronautical and mechanical engineering Finite element models are used widely to model the dynamic behaviour of many systems including in electrical, aerospace and mechanical engineering. The book covers probabilistic finite element model updating, achieved using Bayesian statistics. The Bayesian framework is employed to estimate the probabilistic finite element models which take into account of the uncertainties in the measurements and the modelling procedure. The Bayesian formulation achieves this by formulating the finite element model as the posterior distribution of the model given the measured data within the context of computational statistics and applies these in aeronautical and mechanical engineering. Probabilistic Finite Element Model Updating Using Bayesian Statistics contains simple explanations of computational statistical techniques such as Metropolis-Hastings Algorithm, Slice sampling, Markov Chain Monte Carlo method, hybrid Monte Carlo as well as Shadow Hybrid Monte Carlo and their relevance in engineering. Key features: Contains several contributions in the area of model updating using Bayesian techniques which are useful for graduate students. Explains in detail the use of Bayesian techniques to quantify uncertainties in mechanical structures as well as the use of Markov Chain Monte Carlo techniques to evaluate the Bayesian formulations. The book is essential reading for researchers, practitioners and students in mechanical and aerospace engineering.
Probabilistic Forecasts and Optimal Decisions
by Roman KrzysztofowiczAccount for uncertainties and optimize decision-making with this thorough exposition Decision theory is a body of thought and research seeking to apply a mathematical-logical framework to assessing probability and optimizing decision-making. It has developed robust tools for addressing all major challenges to decision making. Yet the number of variables and uncertainties affecting each decision outcome, many of them beyond the decider’s control, mean that decision-making is far from a ‘solved problem’. The tools created by decision theory remain to be refined and applied to decisions in which uncertainties are prominent. Probabilistic Forecasts and Optimal Decisions introduces a theoretically-grounded methodology for optimizing decision-making under conditions of uncertainty. Beginning with an overview of the basic elements of probability theory and methods for modeling continuous variates, it proceeds to survey the mathematics of both continuous and discrete models, supporting each with key examples. The result is a crucial window into the complex but enormously rewarding world of decision theory. Readers of Probablistic Forecasts and Optimal Decisions will also find: Extended case studies supported with real-world dataMini-projects running through multiple chapters to illustrate different stages of the decision-making processEnd of chapter exercises designed to facilitate student learning Probabilistic Forecasts and Optimal Decisions is ideal for advanced undergraduate and graduate students in the sciences and engineering, as well as predictive analytics and decision analytics professionals.
Probabilistic Graphical Models: Principles and Applications (Advances in Computer Vision and Pattern Recognition)
by Luis Enrique SucarThis fully updated new edition of a uniquely accessible textbook/reference provides a general introduction to probabilistic graphical models (PGMs) from an engineering perspective. It features new material on partially observable Markov decision processes, graphical models, and deep learning, as well as an even greater number of exercises.The book covers the fundamentals for each of the main classes of PGMs, including representation, inference and learning principles, and reviews real-world applications for each type of model. These applications are drawn from a broad range of disciplines, highlighting the many uses of Bayesian classifiers, hidden Markov models, Bayesian networks, dynamic and temporal Bayesian networks, Markov random fields, influence diagrams, and Markov decision processes.Topics and features:Presents a unified framework encompassing all of the main classes of PGMsExplores the fundamental aspects of representation, inference and learning for each techniqueExamines new material on partially observable Markov decision processes, and graphical modelsIncludes a new chapter introducing deep neural networks and their relation with probabilistic graphical models Covers multidimensional Bayesian classifiers, relational graphical models, and causal modelsProvides substantial chapter-ending exercises, suggestions for further reading, and ideas for research or programming projectsDescribes classifiers such as Gaussian Naive Bayes, Circular Chain Classifiers, and Hierarchical Classifiers with Bayesian NetworksOutlines the practical application of the different techniquesSuggests possible course outlines for instructorsThis classroom-tested work is suitable as a textbook for an advanced undergraduate or a graduate course in probabilistic graphical models for students of computer science, engineering, and physics. Professionals wishing to apply probabilistic graphical models in their own field, or interested in the basis of these techniques, will also find the book to be an invaluable reference.Dr. Luis Enrique Sucar is a Senior Research Scientist at the National Institute for Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico. He received the National Science Prize en 2016.
Probabilistic Machine Learning for Civil Engineers
by James-A. GouletAn introduction to key concepts and techniques in probabilistic machine learning for civil engineering students and professionals; with many step-by-step examples, illustrations, and exercises.This book introduces probabilistic machine learning concepts to civil engineering students and professionals, presenting key approaches and techniques in a way that is accessible to readers without a specialized background in statistics or computer science. It presents different methods clearly and directly, through step-by-step examples, illustrations, and exercises. Having mastered the material, readers will be able to understand the more advanced machine learning literature from which this book draws.The book presents key approaches in the three subfields of probabilistic machine learning: supervised learning, unsupervised learning, and reinforcement learning. It first covers the background knowledge required to understand machine learning, including linear algebra and probability theory. It goes on to present Bayesian estimation, which is behind the formulation of both supervised and unsupervised learning methods, and Markov chain Monte Carlo methods, which enable Bayesian estimation in certain complex cases. The book then covers approaches associated with supervised learning, including regression methods and classification methods, and notions associated with unsupervised learning, including clustering, dimensionality reduction, Bayesian networks, state-space models, and model calibration. Finally, the book introduces fundamental concepts of rational decisions in uncertain contexts and rational decision-making in uncertain and sequential contexts. Building on this, the book describes the basics of reinforcement learning, whereby a virtual agent learns how to make optimal decisions through trial and error while interacting with its environment.
Probabilistic Mapping of Spatial Motion Patterns for Mobile Robots (Cognitive Systems Monographs #40)
by Tomasz Piotr Kucner Achim J. Lilienthal Martin Magnusson Luigi Palmieri Chittaranjan Srinivas SwaminathanThis book describes how robots can make sense of motion in their surroundings and use the patterns they observe to blend in better in dynamic environments shared with humans.The world around us is constantly changing. Nonetheless, we can find our way and aren’t overwhelmed by all the buzz, since motion often follows discernible patterns. Just like humans, robots need to understand the patterns behind the dynamics in their surroundings to be able to efficiently operate e.g. in a busy airport. Yet robotic mapping has traditionally been based on the static world assumption, which disregards motion altogether. In this book, the authors describe how robots can instead explicitly learn patterns of dynamic change from observations, store those patterns in Maps of Dynamics (MoDs), and use MoDs to plan less intrusive, safer and more efficient paths. The authors discuss the pros and cons of recently introduced MoDs and approaches to MoD-informed motion planning, and provide an outlook on future work in this emerging, fascinating field.
Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime, and Size Effect
by Zdeněk P. Bažant Jia-Liang LeQuasibrittle (or brittle heterogeneous) materials are becoming increasingly important for modern engineering. They include concretes, rocks, fiber composites, tough ceramics, sea ice, bone, wood, stiff soils, rigid foams, glass, dental and biomaterials, as well as all brittle materials on the micro or nano scale. Their salient feature is that the fracture process zone size is non-negligible compared to the structural dimensions. This causes intricate energetic and statistical size effects and leads to size-dependent probability distribution of strength, transitional between Gaussian and Weibullian. The ensuing difficult challenges for safe design are vanquished in this book, which features a rigorous theory with detailed derivations yet no superfluous mathematical sophistication; extensive experimental verifications; and realistic approximations for design. A wide range of subjects is covered, including probabilistic fracture kinetics at nanoscale, multiscale transition, statistics of structural strength and lifetime, size effect, reliability indices, safety factors, and ramification to gate dielectrics breakdown.
Probabilistic Methods and Distributed Information: Rudolf Ahlswede’s Lectures on Information Theory 5 (Foundations in Signal Processing, Communications and Networking #15)
by Rudolf Ahlswede Alexander Ahlswede Ingo Althöfer Christian Deppe Vladimir Blinovsky Ulrich Tamm Holger Boche Ulrich Krengel Ahmed MansourThe fifth volume of Rudolf Ahlswede’s lectures on Information Theory focuses on several problems that were at the heart of a lot of his research. One of the highlights of the entire lecture note series is surely Part I of this volume on arbitrarily varying channels (AVC), a subject in which Ahlswede was probably the world's leading expert. Appended to Part I is a survey by Holger Boche and Ahmed Mansour on recent results concerning AVC and arbitrarily varying wiretap channels (AVWC). After a short Part II on continuous data compression, Part III, the longest part of the book, is devoted to distributed information. This Part includes discussions on a variety of related topics; among them let us emphasize two which are famously associated with Ahlswede: "multiple descriptions", on which he produced some of the best research worldwide, and "network coding", which had Ahlswede among the authors of its pioneering paper. The final Part IV on "Statistical Inference under Communication constraints" is mainly based on Ahlswede’s joint paper with Imre Csiszar, which received the Best Paper Award of the IEEE Information Theory Society. The lectures presented in this work, which consists of 10 volumes, are suitable for graduate students in Mathematics, and also for those working in Theoretical Computer Science, Physics, and Electrical Engineering with a background in basic Mathematics. The lectures can be used either as the basis for courses or to supplement them in many ways. Ph.D. students will also find research problems, often with conjectures, that offer potential subjects for a thesis. More advanced researchers may find questions which form the basis of entire research programs.
Probabilistic Methods in Geotechnical Engineering: Proceedings of the conference, Canberra, 10-12 February 1993
by K.S.LI and S-C.R.LOThe proceedings of this conference contain keynote addresses on recent developments in geotechnical reliability and limit state design in geotechnics. It also contains invited lectures on such topics as modelling of soil variability, simulation of random fields and probability of rock joints.Contents: Keynote addresses on recent development on geotechnical reliability and limit state design in geotechnics, and invited lectures on modelling of soil variability, simulation of random field, probabilistic of rock joints, and probabilistic design of foundations and slopes. Other papers on analytical techniques in geotechnical reliability, modelling of soil properties, and probabilistic analysis of slopes, embankments and foundations.
Probabilistic Methods in Structural Engineering
by Guiliano Augusti A. Baratta F. CasciatiThis book presents the most important applications of probablistic and statistical approaches and procedures to structural engineering.
Probabilistic Physics of Failure Approach to Reliability: Modeling, Accelerated Testing, Prognosis and Reliability Assessment
by Christopher Jackson Mehdi Amiri Mohammad ModarresThe book presents highly technical approaches to the probabilistic physics of failure analysis and applications to accelerated life and degradation testing to reliability prediction and assessment. Beside reviewing a select set of important failure mechanisms, the book covers basic and advanced methods of performing accelerated life test and accelerated degradation tests and analyzing the test data. The book includes a large number of very useful examples to help readers understand complicated methods described. Finally, MATLAB, R and OpenBUGS computer scripts are provided and discussed to support complex computational probabilistic analyses introduced.
Probabilistic Power System Expansion Planning with Renewable Energy Resources and Energy Storage Systems (IEEE Press Series on Power and Energy Systems)
by Jaeseok Choi Kwang Y. LeeProbabilistic Power System Expansion Planning with Renewable Energy Resources and Energy Storage Systems Discover how modern techniques have shaped complex power system expansion planning with this one-stop resource from two experts in the field Probabilistic Power System Expansion Planning with Renewable Energy Resources and Energy Storage Systems delivers a comprehensive collection of innovative approaches to the probabilistic planning of generation and transmission systems under uncertainties. The book includes renewables and energy storage calculations when using probabilistic and deterministic reliability techniques to assess system performance from a long-term expansion planning viewpoint. Divided into two sections, the book first covers topics related to Generation Expansion Planning, with chapters on cost assessment, methodology and optimization, and more. The second and final section provides information on Transmission System Expansion Planning, with chapters on reliability constraints, probabilistic production cost simulation, and more. Probabilistic Power System Expansion Planning compares the optimization and methodology across dynamic, linear, and integer programming and explores the branch and bound algorithm. Along with case studies to demonstrate how the techniques described within have been applied in complex power system expansion planning problems, readers will enjoy: A thorough discussion of generation expansion planning, including cost assessment, methodology and optimization, and probabilistic production cost An exploration of transmission system expansion planning, including the branch and bound algorithm, probabilistic production cost simulation for TEP, and TEP with reliability constraints An examination of fuzzy decision making applied to transmission system expansion planning A treatment of probabilistic reliability-based grid expansion planning of power systems including wind turbine generators Perfect for power and energy systems designers, planners, operators, consultants, practicing engineers, software developers, and researchers, Probabilistic Power System Expansion Planning with Renewable Energy Resources and Energy Storage Systems will also earn a place in the libraries of practicing engineers who regularly deal with optimization problems.
Probabilistic Reliability Analysis of Power Systems: A Student’s Introduction
by José L. Rueda Torres Francisco M. Gonzalez-Longatt Mart A. van der Meijden Bart W. Tuinema Alexandru I. StefanovThis textbook provides an introduction to probabilistic reliability analysis of power systems. It discusses a range of probabilistic methods used in reliability modelling of power system components, small systems and large systems. It also presents the benefits of probabilistic methods for modelling renewable energy sources. The textbook describes real-life studies, discussing practical examples and providing interesting problems, teaching students the methods in a thorough and hands-on way.The textbook has chapters dedicated to reliability models for components (reliability functions, component life cycle, two-state Markov model, stress-strength model), small systems (reliability networks, Markov models, fault/event tree analysis) and large systems (generation adequacy, state enumeration, Monte-Carlo simulation). Moreover, it contains chapters about probabilistic optimal power flow, the reliability of underground cables and cyber-physical power systems.After reading this book, engineering students will be able to apply various methods to model the reliability of power system components, smaller and larger systems. The textbook will be accessible to power engineering students, as well as students from mathematics, computer science, physics, mechanical engineering, policy & management, and will allow them to apply reliability analysis methods to their own areas of expertise.
Probabilistic Reliability Models
by Igor A. UshakovPractical Approaches to Reliability Theory in Cutting-Edge ApplicationsProbabilistic Reliability Models helps readers understand and properly use statistical methodsand optimal resource allocation to solve engineering problems.The author supplies engineers with a deeper understanding of mathematical models while alsoequipping mathematically oriented readers with a fundamental knowledge of the engineeringrelatedapplications at the center of model building. The book showcases the use of probabilitytheory and mathematical statistics to solve common, real-world reliability problems. Followingan introduction to the topic, subsequent chapters explore key systems and models including:* Unrecoverable objects and recoverable systems* Methods of direct enumeration* Markov models and heuristic models* Performance effectiveness* Time redundancy* System survivability* Aging units and their related systems* Multistate systemsDetailed case studies illustrate the relevance of the discussed methods to real-world technicalprojects including software failure avalanches, gas pipelines with underground storage, andintercontinental ballistic missile (ICBM) control systems. Numerical examples and detailedexplanations accompany each topic, and exercises throughout allow readers to test theircomprehension of the presented material.Probabilistic Reliability Models is an excellent book for statistics, engineering, and operationsresearch courses on applied probability at the upper-undergraduate and graduate levels. Thebook is also a valuable reference for professionals and researchers working in industry whowould like a mathematical review of reliability models and the relevant applications.
Probabilistic Risk and Hazard Assessment: Proceedings of the conference, Newcastle, NSW, Australia, 22-23 September 1993
by R.E.MELCHERS; M.G.STEWARTHighlights the multi-disciplinary nature of probabilistic risk and hazard assessment procedures. Topics covered include: Hazard scenario analyses (e.g. HAZOP, FMEA); probabilistic risk assessments; consequence modelling; structural reliability; human error; uncertainty analyses; and risk assessment. Topics are related to the design, construction & operation of chemical & process plants; nuclear facilities; bridges; buildings; offshore structures; dams.
Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)
by Sebastian Thrun Wolfram Burgard Dieter Fox<p>An introduction to the techniques and algorithms of the newest field in robotics. <p>Probabilistic robotics is a new and growing area in robotics, concerned with perception and control in the face of uncertainty. Building on the field of mathematical statistics, probabilistic robotics endows robots with a new level of robustness in real-world situations. This book introduces the reader to a wealth of techniques and algorithms in the field. All algorithms are based on a single overarching mathematical foundation. Each chapter provides example implementations in pseudo code, detailed mathematical derivations, discussions from a practitioner's perspective, and extensive lists of exercises and class projects. The book's Web site, www.probabilistic-robotics.org, has additional material. The book is relevant for anyone involved in robotic software development and scientific research. It will also be of interest to applied statisticians and engineers dealing with real-world sensor data.</p>
Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)
by Sebastian Thrun Wolfram Burgard Dieter FoxAn introduction to the techniques and algorithms of the newest field in robotics. Probabilistic robotics is a new and growing area in robotics, concerned with perception and control in the face of uncertainty. Building on the field of mathematical statistics, probabilistic robotics endows robots with a new level of robustness in real-world situations. This book introduces the reader to a wealth of techniques and algorithms in the field. All algorithms are based on a single overarching mathematical foundation. Each chapter provides example implementations in pseudo code, detailed mathematical derivations, discussions from a practitioner's perspective, and extensive lists of exercises and class projects. The book's Web site, www.probabilistic-robotics.org, has additional material. The book is relevant for anyone involved in robotic software development and scientific research. It will also be of interest to applied statisticians and engineers dealing with real-world sensor data.
Probabilistic-Statistical Approaches to the Prediction of Aircraft Navigation Systems Condition (Springer Aerospace Technology)
by Eliseev B. P. Kozlov A. I. Romancheva N. I. Shatrakov Y. G. Zatuchny D. A. Zavalishin O. I.This book highlights the development of new methods for assessing and forecasting the state of various complex ageing systems in service; analyzing the influence of destabilizing factors on the accuracy of aircraft flight navigation support; and making recommendations on the ideal aircraft route, taking into consideration the available information on the reliability of the navigation and communication equipment.
Probabilistic-Statistical Methods for Risk Assessment in Civil Aviation (Springer Aerospace Technology)
by Valery Dmitryevich Sharov Vadim Vadimovich Vorobyov Dmitry Alexandrovich ZatuchnyThis book analyses the models for major risks related to flight safety in the aviation sector and presents risk estimation methods through examples of several known aviation enterprises. The book provides a comprehensive content for professionals engaged in the development of flight safety regulatory framework as well as in the design and operation of ground-based or on-board flight support radio electronic systems. The book is also useful for senior students and postgraduates in aviation specialties, especially those related to air traffic management.
Probabilistic Transmission System Planning
by Wenyuan LiThe book is composed of 12 chapters and three appendices, and can be divided into four parts. The first part includes Chapters 2 to 7, which discuss the concepts, models, methods and data in probabilistic transmission planning. The second part, Chapters 8 to 11, addresses four essential issues in probabilistic transmission planning applications using actual utility systems as examples. Chapter 12, as the third part, focuses on a special issue, i.e. how to deal with uncertainty of data in probabilistic transmission planning. The fourth part consists of three appendices, which provide the basic knowledge in mathematics for probabilistic planning.
Probabilità, Statistica e Simulazione: Programmi applicativi scritti in R (UNITEXT #131)
by Alberto Rotondi Paolo Pedroni Antonio PievatoloIl libro contiene in forma compatta il programma svolto negli insegnamenti introduttivi di Statistica e tratta alcuni argomenti indispensabili per l'attività di ricerca, come le tecniche di simulazione Monte Carlo, i metodi di inferenza statistica, di best fit e di analisi dei dati di laboratorio. Gli argomenti vengono sviluppati partendo dai fondamenti, evidenziandone gli aspetti applicativi, fino alla descrizione dettagliata di molti casi di particolare rilevanza in ambito scientifico e tecnico. Il testo è rivolto agli studenti universitari dei corsi ad indirizzo scientifico e a tutti quei ricercatori che devono risolvere problemi concreti che coinvolgono l’analisi dei dati e le tecniche di simulazione. In questa edizione, completamente rivista e corretta, sono stati aggiunti alcuni importanti argomenti sul test d’ipotesi (a cui è stato dedicato un capitolo interamente nuovo) e sul trattamento degli errori sistematici. Per la prima volta è stato adottato il software R, con una ricca libreria di programmi originali accessibile al lettore.
Probability and Random Processes for Electrical and Computer Engineers
by John A. GubnerThe theory of probability is a powerful tool that helps electrical and computer engineers to explain, model, analyze, and design the technology they develop. The text begins at the advanced undergraduate level, assuming only a modest knowledge of probability, and progresses through more complex topics mastered at graduate level. The first five chapters cover the basics of probability and both discrete and continuous random variables. The later chapters have a more specialized coverage, including random vectors, Gaussian random vectors, random processes, Markov Chains, and convergence. Describing tools and results that are used extensively in the field, this is more than a textbook; it is also a reference for researchers working in communications, signal processing, and computer network traffic analysis. With over 300 worked examples, some 800 homework problems, and sections for exam preparation, this is an essential companion for advanced undergraduate and graduate students. Further resources for this title, including solutions (for Instructors only), are available online at www. cambridge. org/9780521864701.
Probability and Random Variables for Electrical Engineering: Probability: Measurement of Uncertainty (Studies in Systems, Decision and Control #390)
by Muammer Catak Tofigh Allahviranloo Witold PedryczThis book delivers a concise and carefully structured introduction to probability and random variables. It aims to build a linkage between the theoretical conceptual topics and the practical applications, especially in the undergraduate engineering area. The book motivates the student to gain full understanding of the fundamentals of probability theory and help acquire working problem-solving skills and apply the theory to engineering applications. Each chapter includes solved examples at varying levels (both introductory and advanced) in addition to problems that demonstrate the relevance of the probability and random variables in engineering. As authors, we focused on to find out the optimum ways in order to introduce the topics in probability and random variables area.