Probability, Random Variables and Stochastic Processes, written by S. Unnikrishna Pillai, provides an introduction to and overview of the analysis of chance, randomness and stochastic phenomena in different areas of science and engineering. Throughout the book, Pillai goes into great detail to discuss important key concepts and methods, including probability theory, random variables, limit theorems, martingales and queueing theory.
The book begins by introducing the fundamental concepts and principles of probability theory and its various applications. It then proceeds to discuss random variables and the various operations performed between them (addition, multiplication, integration and differentiation). The book then moves on to discuss important limit theorems, including the Law of Large Numbers, the Central Limit Theorem and the Ergodic Theorem. These theorems will be invaluable to those interested in statistical inference and sampling theory, which are both important in areas of applied mathematics.
After introducing these fundamentals, Pillai goes into further detail in analyzing and discovering the properties of random variables, including the distributions of particular kinds of random variables, such as continuous and discrete uniform, exponential, Poisson and normal distributions. He also discusses correlation and regression analysis, which are inverse processes that enable the user to discover relationships between two or more variables.
The second half of the book examines the more advanced topics of stochastic processes, which provide models of how systems change over time. Pillai covers a wide range of topics such as Markov chains, random walks and branching processes, as well as Brownian motion, which is fundamental to the modelling of diffusion processes. Martingales are also discussed, which are theorized to explain the ambiguities of stock markets. He then transitions to discuss queueing theory and its applications to the study of production and service systems.
Overall, the book provides a comprehensive treatment on the analysis of chance, randomness and stochastic phenomena. Students and professionals interested in these topics will find this book to be an invaluable resource. Pillai's writing style is both engaging and accessible, making the material easy to digest. Probability, Random Variables and Stochastic Processes is an essential book for anyone working in or looking to pursue a career in applied mathematics.
