Discrete \(q\)-distributions (Q2803796)

This book deals with \(q\)-discrete distributions. It is divided into five sections. The first one is an introductory section and it starts with the definitions of the \(q\)-factorials and \(q\)-binomial coefficients. After introducing the basic concepts, the author gives some expressions and properties of these coefficients. Beside this, some other formulas are considered in details such as \(q\)-Vandermonde, \(q\)-Cauchy, \(q\)-binomial, negative \(q\)-binomial, \(q\)-Stirling numbers, etc. The first section ends with reference notes and 38 very useful exercises. In Section 2, the author introduces some distributions which are obtained in independent Bernoulli trials varying the probability of success. The first introduced distribution is the \(q\)-binomial distribution of the first kind with three parameters \(n\), \(\theta\) and \(q\). The parameter \(n\) represents the number of trials, while the two parameters \(\theta\) and \(q\) affect the probability of success. Further, some generalized distributions such as the negative \(q\)-binomial, the Heine and \(q\)-Stirling distribution are considered and their possible applications are discussed. At the end of this section, 24 exercises are given. Section 3 deals also with independent Bernoulli trials, but now the probability of success varies with the number of successes. In this sense, the author introduces some distributions of the second kind, the Euler and the \(q\)-logarithmic distribution. Some properties of these distributions are derived and some possible applications in real life are discussed. Again, the section ends with a large number of exercises. In Section 4, the probability of success varies with additional requirement, now with the number of trials and the number of successes. The last section is devoted to some limiting distributions. In this section, some distributions are obtained from some limiting processes, for example when the number of trials tends to infinity, or when the number of failures tends to infinity. At the end of this section some central limit theorems are given. To conclude, this book is an interesting and well-written monograph and presents a very useful resource for students and researchers. It contains a large number of exercises and references.