P. Mohana Shankar - Probability, Random Variables, and Data Analytics with Engineering Applications

P. Mohana Shankar

Probability, Random Variables, and Data Analytics with Engineering Applications

1st ed. 2021

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Dedicated to my parents Padmanabharao and Kanakabai who were teachers.

The topics in probability and statistics are concept driven, and textbooks devoted to these require a different mode of presentation than the traditional ones seen in mathematics. The key difference is that the examples must be relevant to the students for whom the course in probability is a curricular requirement. While the students might have been exposed to some elements of probability, the idea of experiments with outcomes being described in statistical terms is novel to them. Thus the introduction of randomness and the notion of a random variable are abstract and require example driven methodology in presentation. Since engineering applications of probability include modeling of outcomes and testing the validity of the models, the book must be thematic with particular emphasis on engineering problems that students may encounter either in school or in their professional world after graduation. This book has been prepared keeping this central theme, namely, application-oriented presentation in terms of examples and exercises. Industrial, commercial, and medical applications involve collection, analysis, interpretation of data, etc., and therefore students must also learn computational techniques. This aspect adds another component to the presentation as well as the choice of examples and exercises.

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 priori, conditional and 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.

The concept of a random variable and its importance in modeling the outcomes of experiments are presented in Chap.. 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.

Chapter , 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 a means to obtain 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.