Fundamentals of probability and statistics for engineers

TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T.T Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T.T Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK Copyright ! 2004 John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England Telephone ( 44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All R ights R eserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, D esigns and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court R oad, London W1T 4LP, UK, without the permission in writing of the Publisher R equests to the Publisher should be addressed to the Permissions D epartment, John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to ( 44) 1243 770620 This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other W iley Editorial Offices John Wiley & Sons Inc., 111 R iver Street, Hoboken, NJ 07030, U SA Jossey-Bass, 989 Market Street, San F rancisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park R oad, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-86813-9 (Cloth) ISBN 0-470-86814-7 (Paper) Typeset in 10/12pt Times from LaTeX files supplied by the author, processed by Integra Software Services, Pvt Ltd, Pondicherry, India Printed and bound in Great Britain by Biddles Ltd, Guildford, Surrey This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production TLFeBOOK To the memory of my parents TLFeBOOK TLFeBOOK Contents PREFACE INTRODUCTION 1.1 Organization of Text 1.2 Probability Tables and Computer Software 1.3 Prerequisites xiii 3 PART A: PROBABILITY AND RANDOM VARIABLES BASIC PROBABILITY CONCEPTS 2.1 Elements of Set Theory 2.1.1 Set Operations 2.2 Sample Space and Probability Measure 2.2.1 Axioms of Probability 2.2.2 Assignment of Probability 2.3 Statistical Independence 2.4 Conditional Probability R eference F urther R eading Problems RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 3.1 R andom Variables 3.2 Probability Distributions 3.2.1 Probability D istribution F unction 3.2.2 Probability M ass F unction for D iscrete R andom Variables 12 13 16 17 20 28 28 28 37 37 39 39 41 TLFeBOOK qgÀ1 qy2 y1 À y2 : …5:71† 1 ˆÀ : À TLFeBOOK 151 F unctions of R andom Variables Hence, Equation (5.67) leads to À1 f Y1 Y2 …y1 ; y2 † ˆ f X1 ‰gÀ1 …y†Šf X2 ‰g2 …y†ŠjJj " # " # À… y1 ‡ y2 †2 À… y1 À y2 †2 exp ˆ exp 4 8   À… y21 ‡ y22 † exp ˆ ; …ÀI; ÀI† < … y1 ; y2 † < …I; I†: …5:72† 4 It is of interest to note that the result given by Equation (5.72) can be written as f Y1 Y2 …y1 ; y2 † ˆ f Y1 …y1 †f Y2 …y2 †; …5:73† where  2 Ày1 f Y1 …y1 † ˆ ; exp 1=2 …4†  2 Ày2 f Y2 …y2 † ˆ exp ; 1=2 …4† ÀI < y1 < I; ÀI < y2 < I; implying that, although Y and Y are both functions of X1 and X , they are independent and identically and normally distributed Ex ample 5.19 Problem: for the same distributions assigned to X and X in Example 5.18, determine the jpdf of Y ˆ (X 12 ‡ X 22 )1/2 and Y ˆ X /X Answer: let us first note that Y takes values only in the positive range Hence, f Y1 Y2 …y1 ; y2 † ˆ 0; y1 < 0: F or y ! 0, the transformation y ˆ g(x) admits two solutions They are: x11 ˆ gÀ1 11 …y† ˆ x12 ˆ gÀ1 12 …y† ˆ y1 y2 …1 ‡ y22 †1=2 y1 …1 ‡ y22 †1=2 ; ; and x21 ˆ gÀ1 21 …y† ˆ Àx11 ; x22 ˆ gÀ1 22 …y† ˆ Àx12 : TLFeBOOK .. .FUNDAMENTALS OF PROBABILITY AND STATISTICS FOR ENGINEERS T.T Soong State University of New York at Buffalo, Buffalo, New York, USA TLFeBOOK TLFeBOOK FUNDAMENTALS OF PROBABILITY AND STATISTICS. .. of Probability and Statistics for Engineers T.T Soong ! 2004 John Wiley & Sons, Ltd ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB) TLFeBOOK 2.1 F undamentals of Probability and Statistics for Engineers. .. F undamentals of Probability and Statistics for Engineers B A B A (a) A ∪ B (b) A ∩ B Figure 2.3 (a) U nion and (b) intersection of sets A and B The intersection or product of A and B, written