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Probability and statistics are as much about intuition and problem solving as they are about theorem proving. Consequently, students can find it very difficult to make a successful transition from lectures to examinations to practice because the problems involved can vary so much in nature. Since the subject is critical in so many applications from insurance to telecommunications to bioinformatics, the authors have collected more than 200 worked examples and examination questions with complete solutions to help students develop a deep understanding of the subject rather than a superficial knowledge of sophisticated theories. With amusing stories and historical asides sprinkled throughout, this enjoyable book will leave students better equipped to solve problems in practice and under exam conditions.
The original motivation for writing this book was rather personal. The first author, in the course of his teaching career in the Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cambridge, and St John's College, Cambridge, had many painful experiences when good (or even brilliant) students, who were interested in the subject of mathematics and its applications and who performed well during their first academic year, stumbled or nearly failed in the exams. This led to great frustration, which was very hard to overcome in subsequent undergraduate years. A conscientious tutor is always sympathetic to such misfortunes, but even pointing out a student's obvious weaknesses (if any) does not always help. For the second author, such experiences were as a parent of a Cambridge University student rather than as a teacher.
We therefore felt that a monograph focusing on Cambridge University mathematics examination questions would be beneficial for a number of students. Given our own research and teaching backgrounds, it was natural for us to select probability and statistics as the overall topic. The obvious starting point was the first-year course in probability and the second-year course in statistics. In order to cover other courses, several further volumes will be needed; for better or worse, we have decided to embark on such a project.
This book is partially based on the material covered in several Cambridge Mathematical Tripos courses: the third-year undergraduate courses Information Theory (which existed and evolved over the last four decades under slightly varied titles) and Coding and Cryptography (a much younger and simplified course avoiding cumbersome technicalities), and a number of more advanced Part III courses (Part III is a Cambridge equivalent to an MSc in Mathematics). The presentation revolves, essentially, around the following core concepts: (a) the entropy of a probability distribution as a measure of ‘uncertainty’ (and the entropy rate of a random process as a measure of ‘variability’ of its sample trajectories), and (b) coding as a means to measure and use redundancy in information generated by the process.
Thus, the contents of this book includes a more or less standard package of information-theoretical material which can be found nowadays in courses taught across the world, mainly at Computer Science and Electrical Engineering Departments and sometimes at Probability and/or Statistics Departments. What makes this book different is, first of all, a wide range of examples (a pattern that we followed from the onset of the series of textbooks Probability and Statistics by Example by the present authors, published by Cambridge University Press). Most of these examples are of a particular level adopted in Cambridge Mathematical Tripos exams. Therefore, our readers can make their own judgement about what level they have reached or want to reach.