Basic Probability Theory for Biomedical Engineers - JohnD. Enderle
ABSTRACT
This is the first 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, engineers and scientists at all levels of background and experience for the
application of this theory to a wide variety of problems—as well as pursue these topics at a more
advanced level. The 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.
Later chapters simply expand upon these key ideas and extend the range of application. 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. Biomedical engineering examples are introduced
throughout the text and a large number of self-study problems are available for the reader.
KEYWORDS
Probability Theory, Random Processes, Engineering Statistics, Probability and Statistics for
Biomedical Engineers, Statistics.
Introduction
We all face uncertainty. A chance encounter, an unpredicted rain or something more serious
such as an auto accident or illness. Ordinarily, the uncertainty faced in our daily routine is never
quantified and is left as a feeling or intuition. In engineering applications, however, uncertainty
must be quantitatively defined, and analyzed in a mathematically rigorous manner resulting in
an appropriate and consistent solution. Probability theory provides the tools to analyze, in a
deductive manner, the nondeterministic or random aspects of a problem. Our goal is to develop
this theory in an axiomatic framework and to demonstrate how it can be used to solve many
practical problems in electrical engineering.
In this first chapter, we introduce the elementary aspects of probability theory upon which
the following chapter on random variables and chapters in subsequent short books are based.
The discussion of probability theory in this book provides a strong foundation for continued
study in virtually every field of biomedical engineering, and many of the techniques developed
may also be applied to other disciplines.
The theory of probability provides procedures for analyzing random phenomena, phenomena
which exhibit behavior that is unpredictable or cannot be determined exactly. Moreover,
understanding probability theory is essential before one can use statistics. An easy way to explain
what is meant by probability theory is to examine several physical situations that lead to
probability theory problems. First consider tossing a fair coin and predicting the outcome of
the toss. It is impossible to exactly predict the outcome of the coin flip, so the most we can
do is state a chance of our prediction occurring. Next, consider telemetry or a communication
system. The signal received consists of the message and/or data plus an undesired signal called
thermal noise which is heard as a hiss. The noise is caused by the thermal or random motion
of electrons in the conducting media of the receiver—wires, resistors, etc. The signal received
also contains noise picked up as the signal travels through the atmosphere. Note that it is impossible
to exactly compute the value of the noise caused by the random motion of the billions
of charged particles in the receiver’s amplification stages or added in the environment. Thus, it
is impossible to completely remove the undesired noise from the signal. We will see, however,
that probability theory provides a means by which most of the unwanted noise is removed.
Download
http://s18.alxa.net/s18/srvs2/02/003...nD.Enderle.rar
ABSTRACT
This is the first 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, engineers and scientists at all levels of background and experience for the
application of this theory to a wide variety of problems—as well as pursue these topics at a more
advanced level. The 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.
Later chapters simply expand upon these key ideas and extend the range of application. 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. Biomedical engineering examples are introduced
throughout the text and a large number of self-study problems are available for the reader.
KEYWORDS
Probability Theory, Random Processes, Engineering Statistics, Probability and Statistics for
Biomedical Engineers, Statistics.
Introduction
We all face uncertainty. A chance encounter, an unpredicted rain or something more serious
such as an auto accident or illness. Ordinarily, the uncertainty faced in our daily routine is never
quantified and is left as a feeling or intuition. In engineering applications, however, uncertainty
must be quantitatively defined, and analyzed in a mathematically rigorous manner resulting in
an appropriate and consistent solution. Probability theory provides the tools to analyze, in a
deductive manner, the nondeterministic or random aspects of a problem. Our goal is to develop
this theory in an axiomatic framework and to demonstrate how it can be used to solve many
practical problems in electrical engineering.
In this first chapter, we introduce the elementary aspects of probability theory upon which
the following chapter on random variables and chapters in subsequent short books are based.
The discussion of probability theory in this book provides a strong foundation for continued
study in virtually every field of biomedical engineering, and many of the techniques developed
may also be applied to other disciplines.
The theory of probability provides procedures for analyzing random phenomena, phenomena
which exhibit behavior that is unpredictable or cannot be determined exactly. Moreover,
understanding probability theory is essential before one can use statistics. An easy way to explain
what is meant by probability theory is to examine several physical situations that lead to
probability theory problems. First consider tossing a fair coin and predicting the outcome of
the toss. It is impossible to exactly predict the outcome of the coin flip, so the most we can
do is state a chance of our prediction occurring. Next, consider telemetry or a communication
system. The signal received consists of the message and/or data plus an undesired signal called
thermal noise which is heard as a hiss. The noise is caused by the thermal or random motion
of electrons in the conducting media of the receiver—wires, resistors, etc. The signal received
also contains noise picked up as the signal travels through the atmosphere. Note that it is impossible
to exactly compute the value of the noise caused by the random motion of the billions
of charged particles in the receiver’s amplification stages or added in the environment. Thus, it
is impossible to completely remove the undesired noise from the signal. We will see, however,
that probability theory provides a means by which most of the unwanted noise is removed.
Download
http://s18.alxa.net/s18/srvs2/02/003...nD.Enderle.rar