Probability and Statistical Inference. Robert Bartoszynski

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СКАЧАТЬ 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, and are likely to remain indefinitely, our best and most reliable prediction tools.

      To make decisions under uncertainty, one usually needs to collect some data. Data may come from past experiences and observations, or may result from some controlled processes, such as laboratory or field experiments. The data are then used to hypothesize about the laws (often called mechanisms) that govern the fragment of reality of interest. In our book, we are interested in laws expressed in probabilistic terms: They specify directly, or allow us to compute, the chances of some events to occur. Knowledge of these chances is, in most cases, the best one can get with regard to prediction and decisions.

Probability theory is a domain of pure mathematics and as such, it has its own conceptual structure. To enable a variety of applications (typically comprising of all areas of human endeavor, ranging from biological, medical, social and physical sciences, to engineering, humanities, business, etc.), such structure must be kept on an abstract level. An application of probability to the particular situation analyzed requires a number of initial steps in which the elements of the real situation are interpreted as abstract concepts of probability theory. Such interpretation is often referred to as building a probabilistic model of the situation at hand. How well this is done is crucial to the success of application.

      One of the main concepts here is that of an experiment—a term used in a broad sense. It means any process that generates data which is influenced, at least in part, by chance.

      

1.2 Sample Space

      In analyzing an experiment, one is primarily interested in its outcome—the concept that is not defined (i.e., a primitive concept) but has to be specified in every particular application. This specification may be done in different ways, with the only requirements being that (1) outcomes exclude one another and (2) they exhaust the set of all logical possibilities.

Example 1.1

Consider an experiment consisting of two tosses of a regular die. An outcome is most СКАЧАТЬ