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xii Preface

familiar to readers who have taken a course in basic probability theory. Chapter 12 requires
Chapter 11 and Chapters 13 and 14 require Chapter 12; in addition, Sections 13.3 and 13.5
use material from Sections 7.5 and 7.6.
The mathematical prerequisites for this book are modest. Good backgrounds in calculus
and linear algebra are important and a course in elementary mathematical analysis at the
level of Rudin (1976) is useful, but not required. Appendix 3 gives a detailed summary of
the mathematical definitions and results that are used in the book.
Although many results from elementary probability theory are presented in Chapters 1
to 4, it is assumed that readers have had some previous exposure to basic probability
theory. Measure theory, however, is not needed and is not used in the book. Thus, although
measurability is briefly discussed in Chapter 1, throughout the book all subsets of a given
sample space are implictly assumed to be measurable. The main drawback of this is that it
is not possible to rigorously define an integral with respect to a distribution function and
to establish commonly used properties of this integral. Although, ideally, readers will have
had previous exposure to integration theory, it is possible to use these results without fully
understanding their proofs; to help in this regard, Appendix 1 contains a brief summary of
the integration theory needed, along with important properties of the integral.
Proofs are given for nearly every result stated. The main exceptions are results requiring
measure theory, although there are surprisingly few results of this type. In these cases,
Ihave tried to outline the basic ideas of the proof and to give an indication of why more
sophisticated mathematical results are needed. The other exceptions are a few cases in which
a proof is given for the case of real-valued random variables and the extension to random
vectors is omitted and a number of cases in which the proof is left as an exercise. I have
not attempted to state results under the weakest possible conditions; on the contrary, I have
often imposed relatively strong conditions if that allows a simpler and more transparent
proof.

Evanston, IL, January, 2005
Thomas A. Severini
severini@northwestern.edu