Advanced Probability Theory for Biomedical Engineers - John D. Enderle
Preface
This is the third in a series of short books on probability theory and random processes for
biomedical engineers. This text is written as an introduction to probability theory. The goal
was to prepare students at the sophomore, junior or senior level for the application of this
theory to a wide variety of problems - as well as pursue these topics at a more advanced
level. Our approach is to present a unified treatment of the subject. There are only a few key
concepts involved in the basic theory of probability theory. These key concepts are all presented
in the first chapter. The second chapter introduces the topic of random variables. The third
chapter focuses on expectation, standard deviation, moments, and the characteristic function.
In addition, conditional expectation, conditional moments and the conditional characteristic
function are also discussed. The fourth chapter introduces jointly distributed random variables,
along with joint expectation, joint moments, and the joint characteristic function. Convolution
is also developed. Later chapters simply expand upon these key ideas and extend the range of
application.
This short book focuses on standard probability distributions commonly encountered in
biomedical engineering. Here in Chapter 5, 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 distributed Gaussian random variables are presented.
The primary subjects of Chapter 6 are methods for determining the 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 variables using
the joint PDF and the CDF.
Aconsiderable 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 applications and examples given reflect the authors’ background in teaching probability
theory and random processes for many years. We have found it best to introduce this
material using simple examples such as dice and cards, rather than more complex biological
and biomedical phenomena. However, we do introduce some pertinent biomedical engineering
examples throughout the text.
Students in other fields should also find the approach useful. Drill problems, straightforward
exercises designed to reinforce concepts and develop problem solution skills, follow most
sections. 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.
We acknowledge and thank William Pruehsner for the technical illustrations. Many of the
examples and end of chapter problems are based on examples from the textbook by Drake [9].
Download
*
Preface
This is the third in a series of short books on probability theory and random processes for
biomedical engineers. This text is written as an introduction to probability theory. The goal
was to prepare students at the sophomore, junior or senior level for the application of this
theory to a wide variety of problems - as well as pursue these topics at a more advanced
level. Our approach is to present a unified treatment of the subject. There are only a few key
concepts involved in the basic theory of probability theory. These key concepts are all presented
in the first chapter. The second chapter introduces the topic of random variables. The third
chapter focuses on expectation, standard deviation, moments, and the characteristic function.
In addition, conditional expectation, conditional moments and the conditional characteristic
function are also discussed. The fourth chapter introduces jointly distributed random variables,
along with joint expectation, joint moments, and the joint characteristic function. Convolution
is also developed. Later chapters simply expand upon these key ideas and extend the range of
application.
This short book focuses on standard probability distributions commonly encountered in
biomedical engineering. Here in Chapter 5, 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 distributed Gaussian random variables are presented.
The primary subjects of Chapter 6 are methods for determining the 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 variables using
the joint PDF and the CDF.
Aconsiderable 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 applications and examples given reflect the authors’ background in teaching probability
theory and random processes for many years. We have found it best to introduce this
material using simple examples such as dice and cards, rather than more complex biological
and biomedical phenomena. However, we do introduce some pertinent biomedical engineering
examples throughout the text.
Students in other fields should also find the approach useful. Drill problems, straightforward
exercises designed to reinforce concepts and develop problem solution skills, follow most
sections. 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.
We acknowledge and thank William Pruehsner for the technical illustrations. Many of the
examples and end of chapter problems are based on examples from the textbook by Drake [9].
Download
*