Introduction to Probability SECOND EDITION Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology
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Course
Probability
Institution
Probability
Introduction to Probability
SECOND EDITION
Dimitri P. Bertsekas and John N. Tsitsiklis
Massachusetts Institute of Technology
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~ Athena Scientific, Belmont, Massachusetts
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Publisher's Cataloging-in-Publication Data
Bertsekas, Dimitri P., Tsitsiklis, John N.
Introduction to Probability
Includes bibliographical references and index
L Probabilities. 2. Stochastic Processes. I. Title.
QA273.B475 2008 519.2 - 21
Library of Congress Control Number: 2002092 167
ISBN 978-1-886529-23-6
,To the memory of
Pantelis Bertsekas and Nikos Tsitsiklis
, Preface
Probability is common sense reduced to calculation
Laplace
This book is an outgrowth of our involvement in teaching an introductory prob
ability course ("Probabilistic Systems Analysis'�) at the Massachusetts Institute
of Technology.
The course is attended by a large number of students with diverse back
grounds, and a broad range of interests. They span the entire spectrum from
freshmen to beginning graduate students, and from the engineering school to the
school of management. Accordingly, we have tried to strike a balance between
simplicity in exposition and sophistication in analytical reasoning. Our key aim
has been to develop the ability to construct and analyze probabilistic models in
a manner that combines intuitive understanding and mathematical precision.
In this spirit, some of the more mathematically rigorous analysis has been
just sketched or intuitively explained in the text. so that complex proofs do not
stand in the way of an otherwise simple exposition. At the same time, some of
this analysis is developed (at the level of advanced calculus) in theoretical prob
lems, that are included at the end of the corresponding chapter. FUrthermore,
some of the subtler mathematical issues are hinted at in footnotes addressed to
the more attentive reader.
The book covers the fundamentals of probability theory (probabilistic mod
els, discrete and continuous random variables, multiple random variables, and
limit theorems) , which are typically part of a first course on the subject. It
also contains, in Chapters 4-6 a number of more advanced topics, from which an
instructor can choose to match the goals of a particular course. In particular, in
Chapter 4, we develop transforms, a more advanced view of conditioning, sums
of random variables, least squares estimation, and the bivariate normal distribu-
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