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Über dieses Buch

This book bridges the gap between theory and applications that currently exist in undergraduate engineering probability textbooks. It offers examples and exercises using data (sets) in addition to traditional analytical and conceptual ones. Conceptual topics such as one and two random variables, transformations, etc. are presented with a focus on applications. Data analytics related portions of the book offer detailed coverage of receiver operating characteristics curves, parametric and nonparametric hypothesis testing, bootstrapping, performance analysis of machine vision and clinical diagnostic systems, and so on. With Excel spreadsheets of data provided, the book offers a balanced mix of traditional topics and data analytics expanding the scope, diversity, and applications of engineering probability. This makes the contents of the book relevant to current and future applications students are likely to encounter in their endeavors after completion of their studies. A full suite of classroom material is included. A solutions manual is available for instructors. Bridges the gap between conceptual topics and data analytics through appropriate examples and exercises; Features 100's of exercises comprising of traditional analytical ones and others based on data sets relevant to machine vision, machine learning and medical diagnostics;Intersperses analytical approaches with computational ones, providing two-level verifications of a majority of examples and exercises.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This chapter offers an overview of the book. The chapter summaries are given along with the relevance of the topics covered.
P. Mohana Shankar

Chapter 2. Sets, Venn Diagrams, Probability, and Bayes’ Rule

Abstract
This chapter begins with the elementary aspects of probability by starting with sets and Venn diagrams. The concepts of probability follow with appropriate descriptions of marginal, joint, conditional, and total probabilities, Bayes’ rule, Bernoulli trials, etc. Examples include those that examine the notion of continuous probability as a prelude to the presentation of random variables in the next chapter. Keeping with the theme of application-oriented content, the chapter contains topics in data analytics such as the estimation of a priori, conditional, and a posteriori probabilities associated with a given set of data collected from measurements. The presentation of the subject matter is organized to offer the reader the importance and relevance of the topics to present-day engineering problems. The association between transition matrix and confusion matrix is introduced to illustrate the connection of the probability concepts to data science. Examples and exercises include conceptual and data analytics-based ones.
P. Mohana Shankar

Chapter 3. Concept of a Random Variable

Abstract
The concept of a random variable and its importance in modeling the outcomes of experiments are presented. While providing traditional coverage of discrete and continuous random variables, densities, distribution functions, and conditional densities, readers are also introduced to mixed random variables as well as mixture densities to reflect the current trends in modeling of data from experiments. The examples of transformation of variables presented offer a peek into the various engineering applications. Unlike other books, data analytic techniques are presented again with the offering of receiver operating characteristic curves (ROC) and performance measures such as the area under the ROC curve (AUC), positive predictive values, etc., thereby relating the subject matter to the topics covered in Chap. 2. Chapter summary provided offers detailed descriptions of densities and their properties. Multiple examples of transformation of variables are also presented. Examples and exercises include traditional analytical ones alongside data-based ones requiring computational approaches.
P. Mohana Shankar

Chapter 4. Multiple Random Variables and Their Characteristics

Abstract
This chapter is devoted to multiple random variables (mainly two variables). The transformation of two variables is presented initially by expanding on the notions of conditional densities in Chap. 3, before invoking the approaches requiring the use of Leibniz theorem and Jacobian. Modeling of outcomes in an experiment is presented as a two-stage experiment. Characteristic functions and Laplace transforms are offered for the determination of the densities of the sum and difference of variables. Mellin transforms (a topic not covered in textbooks) are presented as an approach to finding the densities of the products and ratios of two or more random variables. Meijer G functions are introduced to express the densities of products of random variables. The chapter offers detailed descriptions of the central limit theorem (sums and products) and order statistics. The examples and exercises are applications oriented and often involve the use of computational approaches.
P. Mohana Shankar

Chapter 5. Applications to Data Analytics and Modeling

Abstract
In this chapter is exclusively devoted to data analytics. The topics from previous chapters are invoked to make connections to hypothesis testing (chi-square tests), parameter estimation, ROC, and performance analysis with the aim of developing data analytic approaches. Instead of presenting hypothesis testing and parameter estimation as theoretical topics, they are offered as tools in the context of data analytics, and examples reflect this paradigm shift. ROC analysis is revisited to understand the statistics of the area under the ROC curve through the study of bootstrapping. Examples of bootstrapping are given which examine the method of comparing the areas under the ROC curves generated from data from identical subject pools. A modeling example is presented by examining the statistics of signal fluctuations in wireless channels demonstrating the genesis of several densities. Diversity is introduced as a means to mitigate signal strength fluctuations in wireless channels. All the exercises involve data analytics.
P. Mohana Shankar

Backmatter

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