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Intermediate Probability Theory

for Biomedical Engineers

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.

Intermediate Probability Theory for Biomedical Engineers

John D. Enderle, David C. Farden, and Daniel J. Krause

www.morganclaypool.com

ISBN-10: 1598291408 paperback

ISBN-13: 9781598291407 paperback

ISBN-10: 1598291416 ebook

ISBN-13: 9781598291414 ebook

DOI10.2200/S00062ED1V01Y200610BME010

A lecture in the Morgan & Claypool Synthesis Series

SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING #10

Lecture #10

Series Editor: John D. Enderle, University of Connecticut

Series ISSN: 1930-0328

Series ISSN: 1930-0336

electronic

First Edition

10 9 8 7 6 5 4 3 2 1

Printed in the United States of America

Intermediate 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 #10

& C

Morgan & Claypool Publishers

ABSTRACT

This is the second in a series of three short books on probability theory and random processes for

biomedical engineers. This volume focuses on expectation, standard deviation, moments , and the

characteristic function . In addition, conditional expectation, conditional moments and the conditional

characteristic function are also discussed. Jointly distributed random variables are described, along

with joint expectation, joint moments , and the joint characteristic function. Convolution is also

developed. A considerable effort has been made to develop the theory in a logical manner—

developing special mathematical skills as needed. The mathematical background required of the

reader is basic knowledge of differential calculus. Every effort has been made to be consistent

with commonly used notation and terminology—both within the engineering community as

well as the probability and statistics literature. The aim is to prepare students for the application

of this theory to a wide variety of problems, as well give practicing engineers and researchers a

tool to pursue these topics at a more advanced level. Pertinent biomedical engineering examples

are used throughout the text.

KEYWORDS

Probability Theory, Random Processes, Engineering Statistics, Probability and Statistics for

Biomedical Engineers, Statistics. Biostatistics, Expectation, Standard Deviation, Moments,

Characteristic Function

Contents

Expectation ...................................................................1

3.1 Moments ................................................................2

3.2 Bounds on Probabilities ..................................................10

3.3 Characteristic Function ..................................................14

3.4 Conditional Expectation .................................................23

3.5 Summary ...............................................................25

3.6 Problems ...............................................................26

Bivariate Random Variables ...................................................33

4.1 Bivariate CDF ..........................................................33

4.1.1 Discrete Bivariate Random Variables ...............................39

4.1.2 Bivariate Continuous Random Variables ........................... 43

4.1.3 Bivariate Mixed Random Variables ................................49

4.2 Bivariate Riemann-Stieltjes Integral . . .....................................53

4.3 Expectation .............................................................57

4.3.1 Moments .......................................................58

4.3.2 Inequalities ......................................................62

4.3.3 Joint Characteristic Function . .....................................64

4.4 Convolution ............................................................66

4.5 Conditional Probability ..................................................71

4.6 Conditional Expectation .................................................78

4.7 Summary ...............................................................85

4.8 Problems ...............................................................86

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