A Modern Approach to Probability Theory
Overview This book is intended as a textbook in probability for graduate students in math ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat
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Bert Fristedt
Lawrence Gray
A Modern Approach
to Probability Theory
Springer Science+Business Media, LLC
Bert Fristedt Lawrence Gray School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A.
Library of Congress Cataloging-in-Publication Data Fristedt, Bert, 1937A modem approach to probability theory / Bert Fristedt, Lawrence Gray. p. cm. -- (Probability and its applications) -) and index. Includes bibliographical references (p.
1. Probabilities 1. Gray, Lawrence F. QA273.F92 1996 519.2--dc20
11. Title. III. Series. 96-5687 CIP
Birlchäuser Boston Printed on acid-free paper
© 1997 Springer Science+Business Media New York Ursprünglich erschienen bei Birkhäuser Boston 1997.
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Copyright is not claimed for works of U .S. Government employees. All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhäuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhäuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 978-1-4899-2839-9 ISBN 978-1-4899-2837-5 (eBook) DOI 10.1007/978-1-4899-2837-5
Typeset by the authors in AMSLATEX. 9 8 765 432 1
Contents
List of Tables ........................................................... xv Preface ................................................................. xvi Part I. Probability Spaces, Random Variables, and Expectations Chapter 1. Probability Spaces ......................................... 3 1.1. Introductoryexarnples ............................................. 3 1.2. Ingredients of probability spaces ................................... 6 1.3. a-fields ............................................................ 8 1.4. Borel a-fields ...................................................... 9
Chapter 2. Random Variables ........................................ 11 2.1. Definitions and basic results ...................................... 11 2.2. JRd-valued randorn variables ...................................... 14 2.3. JRoo-valued randorn variables ...................................... 18 2.4. Further exarnples ................................................. 20
Chapter 3. Distribution Functions ................................... 25 3.1. Basic theory ...................................................... 25 3.2. Exarnples of distributions ......................................... 29 3.3. Sorne descriptive terrninology ..................................... 31 3.4. Distributions with densities ....................................... 35 3.5. Further exarnples ...........................................
