Advanced Probability Theory for Biomedical Engineers - John D. Enderle.pdf

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Advanced Probability Theory
for 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 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 paperback
ISBN-13: 9781598291506 paperback
ISBN-10: 1598291513 ebook
ISBN-13: 9781598291513 ebook
DOI 10.2200/S00063ED1V01Y200610BME011
A lecture in the Morgan & Claypool Synthesis Series
SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #11
Lecture #11
Series Editor: John D. Enderle, University of Connecticut
Series ISSN: 1930-0328
print
Series ISSN: 1930-0336
electronic
First Edition
10 9 8 7 6 5 4 3 2 1
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 of Electrical 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
& C
Morgan & Claypool Publishers
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ABSTRACT
This is the third in a series of short books on probability theory and random processes for
biomedical engineers. This book focuses on standard probability distributions commonly en-
countered 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 determining the probability distribution of a func-
tion 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 variables using
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.
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, follow
most sections.
KEYWORDS
Probability Theory, Random Processes, Engineering Statistics, Probability and Statistics for
Biomedical 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 ...............................11
5.3.2 Gaussian Approximation to Bernoulli ..............................12
5.4 Poisson Distribution .....................................................14
5.4.1 Interarrival Times ................................................18
5.5 Univariate Gaussian Distribution . . . ......................................20
5.5.1 Marcum’s Q Function ............................................25
5.6 Bivariate Gaussian Random Variables .....................................26
5.6.1 Constant Contours ...............................................32
5.7 Summary ...............................................................36
5.8 Problems ...............................................................36
6.
Transformations of Random Variables .........................................45
6.1 Univariate CDF Technique .............................................. 45
6.1.1 CDF Technique with Monotonic Functions ........................45
6.1.2 CDF Technique with Arbitrary Functions ..........................46
6.2 Univariate PDF Technique ...............................................53
6.2.1 Continuous Random Variable . ....................................53
6.2.2 Mixed Random Variable . . . .......................................56
6.2.3 Conditional PDF Technique . . ....................................57
6.3 One Function of Two Random Variables .................................. 59
6.4 Bivariate Transformations ................................................63
6.4.1 Bivariate CDF Technique ........................................ 63
6.4.2 Bivariate PDF Technique . . . ......................................65
6.5 Summary ...............................................................73
6.6 Problems ...............................................................75
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