Probability manual

Probability Manual
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Unformatted text preview: THIRD EDITION FUNDAMENTALS OF PROBABILITY WITH STOCHASTIC PROCESSES SAEED GHAHRAMANI Western New England College Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Ghahramani, Saeed. Fundamentals of probability with stochastic processes/ Saeed Ghahramani.3rd edition. p. cm. Includes Index. ISBN: 0-13-145340-8 1. Probabilities. I. Title. QA273.G464 2005 519.2dc22 2004048541 Executive Editor: George Lobell Editor-in-Chief: Sa lly Y agan Production Editor: Jeanne Audino Assistant Managing Editor: Bayani Mendoza DeLeon Senior Managing Editor: Linda Mihatov Behrens Executive Managing Editor: Kathleen Schiaparelli Vice-President/Director of Production and Manufacturing: David W. Riccardi Assistant Manufacturing Manager/Buyer: Michael Bell Manufacturing Manager: Trudy Pisciotti Marketing Manager: Halee Dinsey Marketing Assistant: Rachel Beckman Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Cover Image Specialist: Rita Wenning Cover Photo: Back Cover Photo: Benjamin Shear / Taxi / Getty Images, Inc. Compositor: Saeed Ghahramani Composition: A M S-LAT E X 2005, 2000, 1996 by Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall is a trademark of Pearson Education, Inc. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN 0-13-145340-8 Pearson Education LTD., London Pearson Education of Australia PTY, Limited, Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educacin de Mexico, S.A. de C.V. Pearson Education Japan, Tokyo Pearson Education Malaysia, Pte. Ltd To Lili, Adam, and Andrew C ontents Preface xi 1 Axioms of Probability 1 1.1 Introduction 1 1.2 Sample Space and Events 3 1.3 Axioms of Probability 11 1.4 Basic Theorems 18 1.5 Continuity of Probability Function 27 1.6 Probabilities 0 and 1 29 1.7 Random Selection of Points from Intervals 30 Review Problems 35 2 Combinatorial Methods 38 2.1 Introduction 38 2.2 Counting Principle 38 Number of Subsets of a Set 42 Tree Diagrams 42 2.3 Permutations 47 2.4 Combinations 53 2.5 Stirlings Formula 70 Review Problems 71 3 Conditional Probability and Independence 75 3.1 Conditional Probability 75 Reduction of Sample Space 79 3.2 Law of Multiplication 85 3.3 Law of Total Probability 88 3.4 Bayes Formula 100 3.5 Independence 107 v vi Contents 3.6 Applications of Probability to Genetics 126 Hardy-Weinberg Law 130 Sex-Linked Genes 132 Review Problems 136 4 Distribution Functions and Discrete Random Variables 139 4.1 Random Variables 139 4.2 Distribution Functions 143 4.3 Discrete Random Variables 153 4.4 Expectations of Discrete Random Variables 159 4.5 Variances and Moments of Discrete Random Variables 175 Moments 181 4.6 Standardized Random Variables...
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