[size=120%]An Intermediate Course in Probability (Springer Texts in Statistics)
By Allan Gut

• Publisher:   Springer
• Number Of Pages:   316
• Publication Date:   2009-07-14
• ISBN-10 / ASIN:   1441901612
• ISBN-13 / EAN:   9781441901613

Product Description:

The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of the techniques introduced earlier in the text.
Students are assumed to have taken a first course in probability, though no knowledge of measure theory is assumed. Throughout, the presentation is thorough and includes many examples that are discussed in detail. Thus, students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subject’s many treasures.
The present second edition offers updated content, one hundred additional problems for solution, and a new chapter that provides an outlook on further areas and topics, such as stable distributions and domains of attraction, extreme value theory and records, and martingales. The main idea is that this chapter may serve as an appetizer to the more advanced theory.



Summary: good intermediate level probability text
Rating: 4
This is an intermediate level text in probability theory. It goes beyond the introductory undergraduate course but does not require measure theory like the first (advanced) graduate level course usually does. Sometimes measure theory is required as a co-requisite when taking an advanced course. Gut requires an introduction to probability, calculus and linear algebra as prerequisites. He presents many of the usual topics including important techniques such as generating functions and characteristic functions. So some knowledge of complex variables is also needed.
He introduces the concepts of convergence in distribution, probability, convergence in rth mean and almost sure convergence. Although almost sure convergence is convergence everywhere except on a set with measure zero, Gut carefully avoids this reference to measure theory and simply defines it as
P{on the set of omega that the sequence of random variables Xn converge pointwise to a random variable X} in chapter V1. Up front on the first page in Chapter I Gut introduces the concept of n-dimensional multivariate random variables and defines them as measurable functions mapping from a space of events into Euclidean n-space. He is quick to remark that he will avoid discussing the notion of measurability since measure theory is not a prerequisite. If the student is satisfied with this omission the rest of the book can be read and understood without any knowledge or appreciation of measure theory.

Probability includes a tool box of techniques used to prove limit theorems such as the central limit theorem, the law of large numbers and the law of the iterated logarithm. Gut provides the techniques and a number of exercises and worked out examples to prepare the student for proving limit theorems. The first six chapters cover the standard topics that are usually found in the advanced course including martingales (but no measure theory). Chapter VII is a nice feature of the book. It is an elegant presentation of the Poisson process and some of its generalizations. This gives the student a nice introduction to point processes in general and these processes are used as examples of stochastic processes.

This is also a nice reference for probabilists and statisticians. It contains a nice list of references in the first appendix and solutions to many selected exercises in the third appendix.

Most universities in the US offer the introductory and the advanced course but not an intermediate course such as the one Gut has designed. Perhaps probability could be taught to a wider audience of students in the US if an intermediate course with Gut's book were offered.




Summary: Excellent text
Rating: 5
I bought this text as a supplement to a university course in probability theory. The book is well written with good and relevant examples. I still haven't finished the text but I found that the coverage on items such as transformations, convergence, etc. is very good giving one a strong basis for more theoretical texts.


Summary: nicely written intermediate book
Rating: 4
This is an intermediate level text in probability theory. It goes beyond the introductory undergraduate course but does not require measure theory like the first (advanced) graduate level course usually does. Sometimes measure theory is required as a co-requisite when taking an advanced course. Gut requires an introduction to probability, calculus and linear algebra as prerequisites. He presents many of the usual topics including important techniques such as generating functions and characteristic functions. So some knowledge of complex variables is also needed.
He introduces the concepts of convergence in distribution, probability, convergence in rth mean and almost sure convergence. Although almost sure convergence is convergence everywhere except on a set with measure zero, Gut carefully avoids this reference to measure theory and simply defines it as
P{on the set of omega that the sequence of random variables Xn converge pointwise to a random variable X} in chapter V1. Up front on the first page in Chapter I Gut introduces the concept of n-dimensional multivariate random variables and defines them as measurable functions mapping from a space of events into Euclidean n-space. He is quick to remark that he will avoid discussing the notion of measurability since measure theory is not a prerequisite. If the student is satisfied with this omission the rest of the book can be read and understood without any knowledge or appreciation of measure theory.
Probability includes a tool box of techniques used to prove limit theorems such as the central limit theorem, the law of large numbers and the law of the iterated logarithm. Gut provides the techniques and a number of exercises and worked out examples to prepare the student for proving limit theorems. The first six chapters cover the standard topics that are usually found in the advanced course including martingales (but no measure theory). Chapter VII is a nice feature of the book. It is an elegant presentation of the Poisson process and some of its generalizations. This gives the student a nice introduction to point processes in general and these processes are used as examples of stochastic processes.
This is also a nice reference for probabilists and statisticians. It contains a nice list of references in the first appendix and solutions to many selected exercises in the third appendix.
Most universities in the US offer the introductory and the advanced course but not an intermediate course such as the one Gut has designed. Perhaps probability could be taught to a wider audience of students in the US if an intermediate course with Gut's book were offered.