Probability, Random Variables, and Data Analytics with Engineering Applications

Description

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. Preface Contents Chapter 1: Introduction Chapter 2: Sets, Venn Diagrams, Probability, and Bayes´ Rule 2.1 Introduction 2.2 Algebra of Sets 2.2.1 Summary of Relationships 2.2.2 Overview of Properties of Sets 2.3 Sets, Events, Outcomes, and Probabilities 2.3.1 OR, AND, at Least, and at Most 2.3.2 Probability Expressed as a Continuous Function 2.3.3 Probability as a Geometric Measure 2.4 Conditional Probability and Bayes´ Rule 2.4.1 Conditional Probability, Subsets, and Difference of Two Sets 2.4.2 Conditional Probability and Multiplication Theorem of Probability 2.4.3 Bayes´ Rule of Total Probability: a priori and a posteriori Probabilities 2.4.4 Conditional Probability Revisited: Conditionality and Correlation 2.4.5 Data Analytics, Confusion Matrix, and Positive Predictive Value 2.5 Permutations, Combinations, and Bernoulli Trials 2.6 Summary Exercises References Chapter 3: Concept of a Random Variable 3.1 Introduction 3.2 Distribution and Density Functions 3.2.1 Characteristics of the CDF and pdf 3.2.2 Discrete vs. Continuous Random Variables 3.3 Moments, Characteristic Functions (CHF), Moment Generating Functions (MGF), and Laplace Transforms 3.4 Continuous and Discrete Random Variables: Some Common Densities 3.5 Conditional Densities and Conditional Distribution Functions 3.5.1 Total Probability and Bayes´ Rule 3.5.2 Memoryless Property 3.6 Uniform Density Revisited 3.7 Transformation of Random Variables 3.8 Mixed Random Variables and Mixed Transformations 3.9 Miscellaneous Topics 3.9.1 Hazard Rates and Reliability 3.9.2 Error Rates 3.9.3 Receiver Operating Characteristics Curves 3.9.3.1 ROC Curves Using Probability Density Functions 3.9.3.2 ROC Curves from Measured Data 3.9.3.3 Limitation of the ROC Analysis 3.9.4 Mixture Densities 3.9.5 Some Additional Aspects and Properties of Random Variables 3.9.5.1 Symmetric Random Variables 3.9.5.2 Percentiles 3.9.5.3 Inequalities 3.9.6 Step Functions and Delta Functions 3.9.7 Normal Probability Table (Table 3.2) 3.10 Summary 3.10.1 Overview of Densities and Their Properties 3.10.2 Moment Generating Function and Its Use: Exponential, Gamma, Rayleigh, Uniform, Gaussian, and Poisson Densities 3.10.3 Overview of Transformations 3.10.4 An Alternate View of Transformation of a Random Variable 3.10.5 Conditional Densities and Conditional Expectation Revisited 3.10.6 Proofs of the Moments of Gaussian and Rayleigh Densities 3.10.6.1 Proof of the Gaussian Integral 3.10.6.2 Second Moment of the Gaussian 3.10.6.3 Mean and Second Moment of Rayleigh 3.10.7 Moments of a Poisson Random Variable Exercises References Chapter 4: Multiple Random Variables and Their Characteristics 4.1 Introduction 4.2 Joint Distribution and Densities 4.3 Conditional Densities 4.4 Two-Stage Experiments 4.5 Function of Two Random Variables 4.5.1 Sum of Two Random Variables 4.5.2 Ratio of Two Random Variables 4.5.3 Product of Two Random Variables 4.5.4 Density of g(X,Y) Graphically 4.5.5 Other Useful Transformations: max{X,Y} and min{X,Y} 4.5.6 Joint Density of Two Functions of a Pair of Random Variables 4.6 Joint Moments, Correlation, Covariance, Etc. 4.6.1 Bayes´ Rule, Conditionality and Correlation 4.7 Characteristic Functions, Laplace Transforms, and Mellin Transforms 4.8 Multiple Random Variables 4.8.1 Order Statistics 4.8.2 Central Limit Theorem for the Sum 4.8.3 Central Limit Theorem for the Products 4.8.4 Densities of the Sum of the Squares of Normal Variables 4.9 Summary 4.9.1 Density of a Function of Two or More Independent Variables 4.9.2 Density of the Difference of Two Independent Random Variables (Detailed Analysis) 4.9.3 Leibniz Rule: Differentiation of a Definite Integral Exercises References Chapter 5: Applications to Data Analytics and Modeling 5.1 Introduction 5.2 Receiver Operating Characteristics (ROC) Curves 5.2.1 Optimal Operating Point (OOP) and Positive Predictive Value (PPV) 5.2.2 Statistical Variation of the Area under the ROC Curve 5.2.3 Improvement in Performance and Statistical Modeling 5.2.3.1 Statistics of the Data (Probability Density) Parameter Estimation: Method of Moments (MoM) and Maximum Likelihood Estimation (MLE) 5.2.3.2 Hypothesis Testing: Chi-Square (χ2) Tests 5.2.3.3 Simultaneous Testing of Multiple Hypothesis: A Comprehensive Approach 5.2.3.4 Cautionary Note on Chi-Square Testing (Variability) 5.2.3.5 Additional Data Creation and Analysis 5.2.3.6 ROC Analysis and Performance Enhancement Part 1 ROC Analysis, Theoretical ROC Based on Bigamma Models, Etc. Part 2 Hypothesis Testing and Theoretical ROC Part 3 ROC Analysis, Diversity, and Performance Improvement Part 4 ROC and Diversity Analysis 5.3 Bayesian Decision Theory 5.3.1 ROC Analysis Revisited Through Bayes´ Decision Theory 5.3.2 Optimal Operating Point (Revisited) 5.4 Bootstrapping 5.4.1 Bootstrapping of the Population Mean 5.4.2 Bootstrapping of Area Under the ROC Curve (AUC) 5.4.3 Bootstrapping in Machine Vision and Medical Diagnostics: AUC Comparison 5.5 Applications in Telecommunications 5.5.1 Wireless Channel Models 5.5.1.1 Error Rates and Outage Probabilities 5.5.2 Diversity Techniques 5.6 Applications to Reliability of Interconnected Systems 5.7 Summary Exercises References Appendix A: t-Tests, z-Tests, and p-Values Introduction Data, Samples, Sample Mean, and Sample Variance Confidence Intervals for the Estimation of Mean Interpretation of the Test Statistic p-Value of a Test, z-Tests, and t-Tests Appendix B appendix_book_shankar Find the Sum of a Symbolic Expression I Find the Sum of a Symbolic Expression II Find Sum of Infinite Series in Symbolic Form I Find Sum of Infinite Series in Symbolic Form I Find Sum of Infinite Series in Symbolic Form II Find Sum of Infinite Series in Symbolic Form III Find Sum of Infinite Series in Symbolic Form IV Series Expansions: log(1 + x), -log(1 - x), sin(x), cos(x), Euler´s Identity Differentiation Implicit Differentiation Integration Definite Integrals Definite Integration with More Specifications Symbolic Substitution sin(pi*x)/(pi*x) in the limit x 0 Substitution, Differentiation, Integration, Inline Functions, LaTeX Use Gamma Density: CDF the Ratio of Two Gamma Functions Conditional Density Modeling and Analysis Data Analysis, Extra Data Creation Using Random (.) or Bootstrapping symbolic_MeijerG-use Use of Meijer G Function Numerically MLE and Chi-Square Testing: Generalized Gamma pdf, Not a Built-in pdf Characteristic Function, Laplace and Inverse Laplace, Mellin Bibliography Index