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Basic Probability Theory for Biomedical Engineers - J. Enderle, Et Al., (Morgan and Claypool, 2006) WW

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Basic Probability Theory for Biomedical Engineers

Text of Basic Probability Theory for Biomedical Engineers - J. Enderle, Et Al., (Morgan and Claypool, 2006)...

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    Basic Probability Theoryfor Biomedical Engineers

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  • Copyright 2006 by Morgan & Claypool

    All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted inany form or by any meanselectronic, mechanical, photocopy, recording, or any other except for brief quotationsin printed reviews, without the prior permission of the publisher.

    Basic Probability Theory for Biomedical EngineersJohn D. Enderle, David C. Farden, Daniel J. Krausewww.morganclaypool.com

    ISBN: 1598290606 paperISBN: 9781598290608 paper

    ISBN: 1598290614 ebookISBN: 9781598290615 ebook

    DOI10.2200/S00037ED1V01Y200606BME005Library of Congress Cataloging-in-Publication Data

    A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERINGLecture #5Series Editor and Affliation: John D. Enderle, University of Connecticut

    1930-0328 Print1930-0336 Electronic

    First Edition10 9 8 7 6 5 4 3 2 1

    Printed in the United States of America

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    Basic Probability Theoryfor Biomedical EngineersJohn D. EnderleProgram Director & Professor for Biomedical EngineeringUniversity of Connecticut

    David C. FardenProfessor of Electrical and Computer EngineeringNorth Dakota State University

    Daniel J. KrauseEmeritus Professor of Electrical and Computer EngineeringNorth Dakota State University

    SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #5

    M&C Morgan &Claypool Publishers

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    ABSTRACTThis is the first in a series of short books on probability theory and random processes forbiomedical engineers. This text is written as an introduction to probability theory. The goal wasto prepare students, engineers and scientists at all levels of background and experience for theapplication of this theory to a wide variety of problemsas well as pursue these topics at a moreadvanced level. The approach is to present a unified treatment of the subject. There are onlya few key concepts involved in the basic theory of probability theory. These key concepts areall presented in the first chapter. The second chapter introduces the topic of random variables.Later chapters simply expand upon these key ideas and extend the range of application. Aconsiderable effort has been made to develop the theory in a logical mannerdevelopingspecial mathematical skills as needed. The mathematical background required of the readeris basic knowledge of differential calculus. Every effort has been made to be consistent withcommonly used notation and terminologyboth within the engineering community as wellas the probability and statistics literature. Biomedical engineering examples are introducedthroughout the text and a large number of self-study problems are available for the reader.

    KEYWORDSProbability Theory, Random Processes, Engineering Statistics, Probability and Statistics forBiomedical Engineers, Statistics.

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    Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Preliminary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.2 The Sample Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131.2.1 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141.2.2 Coordinate System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171.2.3 Mathematics of Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    1.3 Definition of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3.1 Classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.3.2 Relative Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.3.3 Personal Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.4 Axiomatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    1.4 The Event Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.5 The Probability Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.6 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.7 Joint Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .481.8 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    2. Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.3 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    2.3.1 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.3.2 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.3.3 Mixed Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    2.4 Riemann-Stieltjes Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1152.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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    PrefaceThis is the first in a series of short books on probability theory and random processes forbiomedical engineers. This text is written as an introduction to probability theory. The goal wasto prepare students at the sophomore, junior or senior level for the application of this theory to awide variety of problemsas well as pursue these topics at a more advanced level. Our approachis to present a unified treatment of the subject. There are only a few key concepts involved in thebasic theory of probability theory. These key concepts are all presented in the first chapter. Thesecond chapter introduces the topic of random variables. Later chapters simply expand uponthese key ideas and extend the range of application.

    A considerable effort has been made to develop the theory in a logical mannerdeveloping special mathematical skills as needed. The mathematical background required of thereader is basic knowledge of differential calculus. Every effort has been made to be consistentwith commonly used notation and terminologyboth within the engineering community aswell as the probability and statistics literature.

    The applications and examples given reflect the authors background in teaching prob-ability theory and random processes for many years. We have found it best to introduce thismaterial using simple examples such as dice and cards, rather than more complex biologicaland biomedical phenomena. However, we do introduce some pertinent biomedical engineeringexamples throughout the text.

    Students in other fields should also find the approach useful. Drill problems, straightfor-ward exercises designed to reinforce concepts and develop problem solution skills, follow mostsections. The answers to the drill problems follow the problem statement in random order.At the end of each chapter is a wide selection of problems, ranging from simple to difficult,presented in the same general order as covered in the textbook.