Applied Probability - Lange K.
Despite the fears of university mathematics departments, mathematics
educat,ion is growing rather than declining. But the truth of the matter
is that the increases are occurring outside departments of mathematics.
Engineers, computer scientists, physicists, chemists, economists, statisti-
cians, biologists, and even philosophers teach and learn a great deal of
mathematics. The teaching is not always terribly rigorous, but it tends to
be better motivated and better adapted to the needs of students. In my
own experience teaching students of biostatistics and mathematical biol-
ogy, I attempt to convey both the beauty and utility of probability. This
is a tall order, partially because probability theory has its own vocabulary
and habits of thought. The axiomatic presentation of advanced probability
typically proceeds via measure theory. This approach has the advantage
of rigor, but it inwitably misses most of the interesting applications, and
many applied scientists rebel against the onslaught of technicalities. In the
current book, I endeavor to achieve a balance between theory and appli-
cations in a rather short compass. While the combination of brevity apd
balance sacrifices many of the proofs of a rigorous course, it is still consis-
tent with supplying students with many of the relevant theoretical tools.
In my opinion, it better to present the mathematical facts without proof
rather than omit them altogether.
In the preface to his lovely recent textbook (1531, David Williams writes,
“Probability and Statistics used to be married; then they separated, then
they got divorced; now they hardly see each other.” Although this split
is doubtless irreversible, at least we ought to be concerned with properly
bringing up their children, applied probability and computational statis-
tics. If we fail, then science as a whole will suffer. You see before you my
attempt to give applied probability the attention it deserves. My other re-
cent book (951 covers computational statistics and aspects of computational
probability glossed over here.
This graduate-level textbook presupposes knowledge of multivariate cal-
culus, linear algehra, and ordinary differential equations. In probability
theory, students should be comfortable with elementary combinatorics, gen-
erating functions, probability densities and distributions, expectations, and
conditioning arguments. My intended audience includes graduate students
in applied mathematics, biostatistics, computational biology, computer sci-
ence, physics, and statistics. Because of the diversity of needs, instructors
are encouraged to exercise their own judgment in deciding what chapters
and.topics to cover.
Chapter 1 reviews elementary probability while striving to give a brief
survey of relevant results from measure theory. Poorly prepared students
should supplement this material with outside reading. Well-prepared stu-
dents can skim Chapter 1 until they reach the less well-knom' material of
the final two sections. Section 1.8 develops properties of the multivariate
normal distribution of special interest to students in biostatistics and sta-
tistics. This material h applied to optimization theory in Section 3.3 and
to diffusion processes in Chapter 11.
We get down to serious business in Chapter 2, which is an extended essay
on calculating expectations. Students often camplain that probability is
nothing more than a bag of tricks. For better or worse, they are confronted
here with some of those tricks. Readers may want to skip the ha1 two
sections of the chapter on surface area distributions on a first pass through
the book.
Chapter 3 touches on advanced topics from convexity, inequalities, and
optimization. Beside the obvious applications to computational statistics,
part of the motivation for this material is its applicability in calculating
bounds on probabilities and moments.
Combinatorics has the odd reputation of being difficult in spite of rely-
ing on elementary methods. Chapters 4 and 5 are my stab at making the
subject accessible and interesting. There is no doubt in my mind of combi-
natorics' practical importance. More and more we live in a world domiuated
by discrete bits of information. The stress on algorithms in Chapter 5 is
intended to appeal to computer scientists.
Chapt,ers 6 through 11 cover core material on stochastic processes that
I have taught to students in mathematical biology over a span of many
years. If supplemented with appropriate sections from Chapters 1 and 2,
there is su6cient material here for a traditional semester-long course in
stochastic processes. Although my examples are weighted toward biology,
particularly genetics, I have tried to achieve variety. The fortunes of this
hook doubtless will hinge on how cornpelling readers find these example.
- Hardcover: 320 pages
- Publisher: Springer (March 12, 2003)
- Language: English
- ISBN-10: 0387004254
- ISBN-13: 978-0387004259
- Product Dimensions: 9.2 x 0.8 x 6.1 inches
Despite the fears of university mathematics departments, mathematics
educat,ion is growing rather than declining. But the truth of the matter
is that the increases are occurring outside departments of mathematics.
Engineers, computer scientists, physicists, chemists, economists, statisti-
cians, biologists, and even philosophers teach and learn a great deal of
mathematics. The teaching is not always terribly rigorous, but it tends to
be better motivated and better adapted to the needs of students. In my
own experience teaching students of biostatistics and mathematical biol-
ogy, I attempt to convey both the beauty and utility of probability. This
is a tall order, partially because probability theory has its own vocabulary
and habits of thought. The axiomatic presentation of advanced probability
typically proceeds via measure theory. This approach has the advantage
of rigor, but it inwitably misses most of the interesting applications, and
many applied scientists rebel against the onslaught of technicalities. In the
current book, I endeavor to achieve a balance between theory and appli-
cations in a rather short compass. While the combination of brevity apd
balance sacrifices many of the proofs of a rigorous course, it is still consis-
tent with supplying students with many of the relevant theoretical tools.
In my opinion, it better to present the mathematical facts without proof
rather than omit them altogether.
In the preface to his lovely recent textbook (1531, David Williams writes,
“Probability and Statistics used to be married; then they separated, then
they got divorced; now they hardly see each other.” Although this split
is doubtless irreversible, at least we ought to be concerned with properly
bringing up their children, applied probability and computational statis-
tics. If we fail, then science as a whole will suffer. You see before you my
attempt to give applied probability the attention it deserves. My other re-
cent book (951 covers computational statistics and aspects of computational
probability glossed over here.
This graduate-level textbook presupposes knowledge of multivariate cal-
culus, linear algehra, and ordinary differential equations. In probability
theory, students should be comfortable with elementary combinatorics, gen-
erating functions, probability densities and distributions, expectations, and
conditioning arguments. My intended audience includes graduate students
in applied mathematics, biostatistics, computational biology, computer sci-
ence, physics, and statistics. Because of the diversity of needs, instructors
are encouraged to exercise their own judgment in deciding what chapters
and.topics to cover.
Chapter 1 reviews elementary probability while striving to give a brief
survey of relevant results from measure theory. Poorly prepared students
should supplement this material with outside reading. Well-prepared stu-
dents can skim Chapter 1 until they reach the less well-knom' material of
the final two sections. Section 1.8 develops properties of the multivariate
normal distribution of special interest to students in biostatistics and sta-
tistics. This material h applied to optimization theory in Section 3.3 and
to diffusion processes in Chapter 11.
We get down to serious business in Chapter 2, which is an extended essay
on calculating expectations. Students often camplain that probability is
nothing more than a bag of tricks. For better or worse, they are confronted
here with some of those tricks. Readers may want to skip the ha1 two
sections of the chapter on surface area distributions on a first pass through
the book.
Chapter 3 touches on advanced topics from convexity, inequalities, and
optimization. Beside the obvious applications to computational statistics,
part of the motivation for this material is its applicability in calculating
bounds on probabilities and moments.
Combinatorics has the odd reputation of being difficult in spite of rely-
ing on elementary methods. Chapters 4 and 5 are my stab at making the
subject accessible and interesting. There is no doubt in my mind of combi-
natorics' practical importance. More and more we live in a world domiuated
by discrete bits of information. The stress on algorithms in Chapter 5 is
intended to appeal to computer scientists.
Chapt,ers 6 through 11 cover core material on stochastic processes that
I have taught to students in mathematical biology over a span of many
years. If supplemented with appropriate sections from Chapters 1 and 2,
there is su6cient material here for a traditional semester-long course in
stochastic processes. Although my examples are weighted toward biology,
particularly genetics, I have tried to achieve variety. The fortunes of this
hook doubtless will hinge on how cornpelling readers find these example.
