Probability and Statistical Inference. Robert Bartoszynski
for seniors or first‐year graduate students in statistics, mathematics, natural sciences, engineering, and any other major where an intensive exposure to statistics is necessary. The prerequisite is a calculus sequence that includes multivariate calculus. We provide the material for a two‐semester course that starts with the necessary background in probability theory, followed by the theory of statistics.
What distinguishes our book from other texts is the way the material is presented and the aspects that are stressed. To put it succinctly, understanding “why” is prioritized over the skill of “how to.” Today, in an era of undreamed‐of computational facilities, a reflection in an attempt to understand is not a luxury but a necessity.
Probability theory and statistics are presented as self‐contained conceptual structures. Their value as a means of description and inference about real‐life situations lies precisely in their level of abstraction—the more abstract a concept is, the wider is its applicability. The methodology of statistics comes out most clearly if it is introduced as an abstract system illustrated by a variety of real‐life applications, not confined to any single domain.
Depending on the level of the course, the instructor can select topics and examples, both in the theory and in applications. These can range from simple illustrations of concepts, to introductions of whole theories typically not included in comparable textbooks (e.g., prediction, extrapolation, and filtration in time series as examples of use of the concepts of covariance and correlation). Such additional, more advanced, material (e.g., Chapter 5 on Markov Chains) is marked with asterisks. Other examples are the proof of the extension theorem (Theorem 6.2.4), showing that the cumulative distribution function determines the measure on the line; the construction of Lebesgue, Riemann–Stieltjes, and Lebesgue–Stieltjes integrals; and the explanation of the difference between double integral and iterated integrals (Section 8.3).
In the material that is seldom included in other textbooks on mathematical statistics, we stress the consequences of nonuniqueness of a sample space and illustrate, by examples, how the choice of a sample space can facilitate the formulation of some problems (e.g., issues of selection or randomized response). We introduce the concept of conditioning with respect to partition (Section 4.4); we explain the Borel–Kolmogorov paradox by way of the underlying measurement process that provides information on the occurrence of the condition (Example 7.22); we present the Neyman–Scott theory of outliers (Example 10.4); we give a new version of the proof of the relation between mean, median, and standard deviation (Theorem 8.7.3); we show another way of conditioning in the secretary problem (Example 4.10). Among examples of applications, we discuss the strategies of serves in tennis (Problem 4.2.12), and a series of problems (3.2.14–3.2.20) concerning combinatorial analysis of voting power. In Chapter 11, we discuss the renewal paradox, the effects of importance sampling (Example 11.6), and the relevance of measurement theory for statistics (Section 11.6). Chapter 14 provides a discussion of true regression versus linear regression and concentrates mostly on explaining why certain procedures (in regression analysis and ANOVA) work, rather than on computational details. In Chapter 15, we provide a taste of rank methods—one line of research starting with the Glivenko–Cantelli Theorem and leading to Kolmogorov–Smirnov tests, and the other line leading to Mann‐Whitney and Wilcoxon tests. In this chapter, we also show the traps associated with multiple tests of the same hypothesis (Example 15.3). Finally, Chapter 16 contains information on partitioning contingency tables—the method that provides insight into the dependence structure. We also introduce McNemar's test and various indices of association for tables with ordered categories.
The backbone of the book is the examples used to illustrate concepts, theorems, and methods. Some examples raise the possibilities of extensions and generalizations, and some simply point out the relevant subtleties.
Another feature that distinguishes our book from most other texts is the choice of problems. Our strategy was to integrate the knowledge students acquired thus far, rather than to train them in a single skill or concept. The solution to a problem in a given section may require using knowledge from some preceding sections, that is, reaching back into material already covered. Most of the problems are intended to make the students aware of facts they might otherwise overlook. Many of the problems were inspired by our teaching experience and familiarity with students' typical errors and misconceptions.
Finally, we hope that our book will be “friendly” for students at all levels. We provide (hopefully) lucid and convincing explanations and motivations, pointing out the difficulties and pitfalls of arguments. We also do not want good students to be left alone. The material in starred chapters, sections, and examples can be skipped in the main part of the course, but used at will by interested students to complement and enhance their knowledge. The book can also be a useful reference, or source of examples and problems, for instructors who teach courses from other texts.
I am indebted to many people without whom this book would not have reached its current form. First, thank you to many colleagues who contributed to the first edition and whose names are listed there. Comments of many instructors and students who used the first edition were influential in this revision. I wish to express my gratitude to Samuel Kotz for referring me to Stigler's (1986) article about the “right and lawful rood,” which we previously used in the book without reference (Example 8.40). My sincere thanks are due to Jung Chao Wang for his help in creating the R‐code for computer‐intensive procedures that, together with additional examples, can be found on the book's ftp site
ftp://ftp.wiley.com/public/sc_tech_med/probability_statistical.
Particular thanks are due to Katarzyna Bugaj for careful proofreading of the entire manuscript, Łukasz Bugaj for meticulously creating all figures with the Mathematica software, and Agata Bugaj for her help in compiling the index. Changing all those diapers has finally paid off.
I wish to express my appreciation to the anonymous reviewers for supporting the book and providing valuable suggestions, and to Steve Quigley, Executive Editor of John Wiley & Sons, for all his help and guidance in carrying out the revision.
Finally, I would like to thank my family, especially my husband Jerzy, for their encouragement and support during the years this book was being written.
Magdalena Niewiadomska‐Bugaj
October 2007
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/probabilityandstatisticalinference3e
The website includes the Instructor's Solution Manual and will be live in early 2021.
Chapter 1 Experiments, Sample Spaces, and Events
1.1 Introduction
The consequences of making a decision today often depend on what will happen in the future, at least on the future that is relevant to the decision. The main purpose of using statistical methods is to help in making better decisions under uncertainty.
Judging from the failures of weather forecasts, to more spectacular prediction failures, such as bankruptcies of large companies and stock market crashes, it would appear that statistical methods do not perform very well. However, with a possible exception of weather forecasting, these examples are, at best, only partially statistical predictions. Moreover, failures tend to be better remembered than successes. Whatever the case, statistical methods are at present,