Advanced Probability Theory For Biomedical Engineers - John D. Enderle

Advanced Probability Theory for Biomedical Engineers

Copyright © 2006 by Morgan & Claypool

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher.Advanced Probability Theory for Biomedical Engineers John D. Enderle, David C. Farden, and Daniel J. Krause www.morganclaypool.com ISBN-10: 1598291505 ISBN-13: 9781598291506 ISBN-10: 1598291513 ISBN-13: 9781598291513 paperback paperback ebook ebook

DOI 10.2200/S00063ED1V01Y200610BME011 A lecture in the Morgan & Claypool Synthesis Series SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #11 Lecture #11Series Editor: John D. Enderle, University of Connecticut

Series ISSN: 1930-0328 Series ISSN: 1930-0336 First Edition 10 9 8 7 6 5 4 3 2 1

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Printed in the United States of America

Advanced Probability Theory for Biomedical Engineers
John D. Enderle
Program Director & Professor for Biomedical Engineering, University of Connecticut

David C. Farden
Professor ofElectrical and Computer Engineering, North Dakota State University

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

SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #11

M &C

Mor gan

& Cl aypool

Publishers

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ABSTRACT
This is the third in a series of short books on probability theory and random processes for biomedicalengineers. This book focuses on standard probability distributions commonly encountered in biomedical engineering. The exponential, Poisson and Gaussian distributions are introduced, as well as important approximations to the Bernoulli PMF and Gaussian CDF. Many important properties of jointly Gaussian random variables are presented. The primary subjects of the final chapter are methods for determiningthe probability distribution of a function of a random variable. We first evaluate the probability distribution of a function of one random variable using the CDF and then the PDF. Next, the probability distribution for a single 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 variablesusing the joint PDF and the CDF. The aim of all three books is as an introduction to probability theory. The audience includes students, engineers and researchers presenting applications of this theory to a wide variety of problems—as well as pursuing these topics at a more advanced level. The theory material is presented in a logical manner—developing special mathematical skills as needed. Themathematical background required of the reader is basic knowledge of differential calculus. Pertinent biomedical engineering examples are throughout the text. Drill problems, straightforward exercises designed to reinforce concepts and develop problem solution skills, follow most sections.

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

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Contents
5. Standard Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5.1 Uniform Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 1 5.2 Exponential Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5.3 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.3.1 Poisson Approximation to Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....