Probability. With applications and R (Q2873692)

The book presents a general introduction to the theory of probability and it can be used as a textbook for the students at the undergraduate level. The book is accompanied by many real-life examples and simulations performed in R. It is a good starting point to learn the theory of probability not only for the students majoring in mathematics, but also in other fields of science, such as biology, physics, computer science, and economics.NEWLINENEWLINEThe first chapter of the book provides basics and general principles of the theory of probability, including the notions of random experiment, sample space, and random event. The multiplication principle and permutations are introduced and random variables are defined. Finally, a first look at simulation is presented.NEWLINENEWLINEChapter 2 discusses conditional probability and presents the related theoretical results, such as the law of total probability and the Bayes formula. In Chapter~3, the notion of independence is introduced and sequences of independent random variables are discussed. Several examples are provided and discrete distributions, such as binomial, Poisson and uniform, are discussed.NEWLINENEWLINEChapter 4 focuses on discrete random variables. Also, joint discrete distributions are introduced. Functions in random variables as well as moments of random variables, such as expectation, variance, covariance and correlation, are discussed in this chapter, while Chapter~5 presents several families of discrete distributions, namely geometric, negative binomial, hypergeometric, multinomial, and Benford's law.NEWLINENEWLINEContinuous random variables are discussed in Chapter~6. Functions of random variables, such as maxima, minima, and sums of independent random variables, are emphasized. Chapter~7 highlights several important continuous distributions starting with the normal distribution. The gamma and beta distributions are presented as well. There is also a section on the Pareto distribution with discussions of power law and scale invariant distributions.NEWLINENEWLINEChapter 8 presents conditional distributions, both in the discrete and continuous settings. Conditional expectation and variance are introduced here as well. Chapter~9 deals with important limit theorems of probability, e.g., the law of large numbers and the central limit theorem. A~part of this section is also devoted to the moment-generating functions. Finally, Chapter~10 provides optional material for supplementary discussions and projects.