Advanced Probability Theory for Biomedical Engineers - John D. Enderle

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    Advanced 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.

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

    ISBN-10: 1598291505 paperbackISBN-13: 9781598291506 paperback

    ISBN-10: 1598291513 ebookISBN-13: 9781598291513 ebook

    DOI 10.2200/S00063ED1V01Y200610BME011

    A lecture in the Morgan & Claypool Synthesis SeriesSYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #11

    Lecture #11Series Editor: John D. Enderle, University of Connecticut

    Series ISSN: 1930-0328 print

    Series ISSN: 1930-0336 electronic

    First Edition10 9 8 7 6 5 4 3 2 1

    Printed in the United States of America

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    Advanced Probability Theoryfor Biomedical EngineersJohn D. EnderleProgram Director & Professor for Biomedical Engineering,University of Connecticut

    David C. FardenProfessor of Electrical and Computer Engineering,North Dakota State University

    Daniel J. KrauseEmeritus Professor of Electrical and Computer Engineering,North Dakota State University

    SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #11

    M&C Morgan &Claypool Publishers

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    ABSTRACTThis is the third in a series of short books on probability theory and random processes forbiomedical engineers. This book focuses on standard probability distributions commonly en-countered in biomedical engineering. The exponential, Poisson and Gaussian distributions areintroduced, as well as important approximations to the Bernoulli PMF and Gaussian CDF.Many important properties of jointly Gaussian random variables are presented. The primarysubjects of the final chapter are methods for determining the probability distribution of a func-tion of a random variable. We first evaluate the probability distribution of a function of onerandom variable using the CDF and then the PDF. Next, the probability distribution for asingle random variable is determined from a function of two random variables using the CDF.Then, the joint probability distribution is found from a function of two random variables usingthe joint PDF and the CDF.

    The aim of all three books is as an introduction to probability theory. The audienceincludes students, engineers and researchers presenting applications of this theory to a widevariety of problemsas well as pursuing these topics at a more advanced level. The theorymaterial is presented in a logical mannerdeveloping special mathematical skills as needed.The mathematical background required of the reader is basic knowledge of differential calculus.Pertinent biomedical engineering examples are throughout the text. Drill problems, straight-forward exercises designed to reinforce concepts and develop problem solution skills, followmost sections.

    KEYWORDSProbability Theory, Random Processes, Engineering Statistics, Probability and Statistics forBiomedical Engineers, Exponential distributions, Poisson distributions, Gaussian distributionsBernoulli PMF and Gaussian CDF. Gaussian random variables

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    Contents5. Standard Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    5.1 Uniform Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Exponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.3 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    5.3.1 Poisson Approximation to Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3.2 Gaussian Approximation to Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    5.4 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.1 Interarrival Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

    5.5 Univariate Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5.1 Marcums Q Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    5.6 Bivariate Gaussian Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6.1 Constant Contours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

    5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    6. Transformations of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1 Univariate CDF Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    6.1.1 CDF Technique with Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . . . 456.1.2 CDF Technique with Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    6.2 Univariate PDF Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.1 Continuous Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.2 Mixed Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2.3 Conditional PDF Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    6.3 One Function of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4 Bivariate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    6.4.1 Bivariate CDF Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.2 Bivariate PDF Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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    PrefaceThis is the third 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 goalwas to prepare students at the sophomore, junior or senior level for the application of thistheory to a wide variety of problems - as well as pursue these topics at a more advancedlevel. Our approach is to present a unified treatment of the subject. There are only a few keyconcepts involved in the basic theory of probability theory. These key concepts are all presentedin the first chapter. The second chapter introduces the topic of random variables. The thirdchapter focuses on expectation, standard deviation, moments, and the characteristic function.In addition, conditional expectation, conditional moments and the conditional characteristicfunction are also discussed. The fourth chapter introduces jointly distributed random variables,along with joint expectation, joint moments, and the joint characteristic function. Convolutionis also developed. Later chapters simply expand upon these key ideas and extend the range ofapplication.

    This short book focuses on standard probability distributions commonly encountered inbiomedical engineering. Here in Chapter 5, the exponential, Poisson and Gaussian distributionsare introduced, as well as important approximations to the Bernoulli PMF and Gaussian CDF.Many important properties of jointly distributed Gaussian random variables are presented.The primary subjects of Chapter 6 are methods for determining the probability distribution ofa function of a random variable. We first evaluate the probability distribution of a function ofone random variable using the CDF and then the PDF. Next, the probability distribution for asingle random variable is determined from a function of two random variables using the CDF.Then, the joint probability distribution is found from a function of two random variables usingthe joint PDF and the CDF.

    A considerable effort has been made to develop the theory in a logical manner - developingspecial mathematical skills as needed. The mathematical background required of the reader isbasic knowledge of differential calculus. Every effort has been made to be consistent withcommonly used notation and terminologyboth within the enginee

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