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Fall 2000
Class meets: Mon, Tues, Thurs, Fri: 3--3:50 p.m. in RAC 113
Professor: Dr. Kevin Iga
Phone number: 456-4313 (office), (818) 880-9439 (home)
Email: kiga@pepperdine.edu
Office: RAC 117
Office hours: Mon., Tues., Thurs. 11-11:50 a.m.
Please come by my office hours whenever you have questions, and even if you
don't have questions. I am also available at other times by appointment.
You are also required to come by my office at one time during the first
or second week of the semester. A signup will be passed around the class.
Texts: Pfeiffer, ``Concepts of Probability Theory'', 2nd ed.
Freund, ``Introduction to Probability'', 1st ed.
The Freund text is somewhat more elementary and we will begin with it, and
much of your homework will be taken from it at first. We will later take a
more serious look at the deep issues in probability through Pfeiffer, and
gradually more readings will be from Pfeiffer.
Unlike many other math courses, the main source of teaching is the
book, as a source of ``real mathematics''. The lectures will seek to
help you through the book. Like many other math courses, the main
source of learning is the homework.
Calculator: A calculator is not required for this course, but you
are allowed to use one for homework and exams.
Prerequisites: Math 212 and either Math 360 or COSC 360.
Other computer resources: A web page and email list will be set
up shortly. You will be informed of its address when that happens.
Objectives: The student should be able:
- To compute various standard probabilities;
- To prove general statements about probabilities, events, and
random variables;
- To compute means and standard deviations of continuous and discrete
probabilities;
- To compute conditional probabilities and to use Bayes' theorem;
- To explain the concepts of measure and its relation to integration;
- To use cumulative distribution functions, probability
density functions, joint cumulative distribution functions, and
joint probability density function;
- To prove the Law of Large Numbers and the Central Limit Theorem;
- To find the stable distribution of simple Markov processes.
Goals: The student should develop:
- An intuition for probability;
- A sense for how to approach mathematical questions involving randomness;
- An appreciation for the philosophical and theoretical issues raised by
probability theory and randomness;
- An increased ability to do mathematical proofs;
- An appreciation for the beauty and practicality of mathematics.
Overview
Probability is a subject that can be studied at several levels. At
first, it seems very simple in theory, although the computations might
be either tedious or tricky. Then, paradoxes will arise that will
require that we reformulate probability more carefully, giving rise to
the axiomatic description. Soon, the need to deal with other
paradoxes gives us a measure-theoretic formulation. As a result, we
will go through the material a few times, each time looking at issues
more deeply. We will then have to spend some time considering if what
results is what we thought we wanted from the beginning.
For the first three weeks, we will take a naive view of probability, as
expressed in Freund's book. In this view, the main difficulty is finding
clever ways to count outcomes, giving us the need (and excuse) to take a
tour of combinatorics.
From September 18, we will think of probability as coming from axioms.
For this, you will need to remember your set theory from math 360/220.
From October 2 through 12, we will be looking at particular situations
and examples that frequently arise in practice, and for this there will
be some memorization. More important is learning to recognize these
situations in real problems. These are ``probability distributions'' and
correspond to random variables.
The introduction of continuous random variables will give us a reason
to consider some paradoxes that occur, giving rise, eventually, to
sets that have no probability. This requires that we be more careful
than before and define the concept of a {\em measure}. The
measure-theoretic approach is the one described in Pfeiffer, and for
this reason, we will begin to concentrate more on Pfeiffer for the
rest of the course. Measure theory and integration theory will occupy
us from October 13 through 27 or so.
Starting October 30, we will explore expected values, variances, and
other properties of random variables. We will also consider relations
between two or more random variables, which leads naturally to the
concepts of correlation, covariance, and independence.
From November 21 through November 30 we will consider the law of large
numbers (which allows us to relate our more sophisticated notion of
probability to a more naive notion) and the central limit theorem, which
explains why the normal distribution is common.
In the last week we will cover a few topics like Markov chains, entropy,
coding theory, and information theory, as time permits.
Homework: Homework will be assigned twice a week: homework assigned
on Tuesday will be due on Friday, and homework assigned on Friday will be due
on Tuesday. Homework should be turned in at the beginning of class.
Remember that the primary place where learning happens is in the
homework, so take the homework seriously. The lowest three homework scores
will be dropped, but you should do your best on all your homework assignments.
Late assignments: You must give me notice that you are going to
turn in an assignment late the class before the assignment is due, or
it will not be accepted. You must also have a good reason. These
reasons will be treated on a case-by-case basis. When you obtain
permission to turn in an assignment late, we will discuss a new due
date for that homework.
Collaboration: You are encouraged to collaborate on all homework
assignments, unless otherwise specified. This means you work on it
independently before discussing it with each other, and it means you
must thoroughly understand how to do the problem before writing it up.
You must write up your answers separately; you cannot turn in one
homework for more than one person, nor can you simply include
photocopies of other students' work. There is no limit to the size
of a group for collaboration, although 3-5 people tends to be an efficient
size.
You should also use these groups to ask questions of each other to
better understand the material. If you do not see each other
frequently, you should set up a regular time and place to meet to work
on assignments. If you do not have a group, talk to me and I can
place you in a group. If you do not wish to work in a group, that is
your prerogative but this will be a disadvantage to you.
Comments: You should include comments about the class at the top of your
homework assignments. These comments can be ``You go too fast'',
``You say `um' too often'', ``I like this chapter'', ``This is too easy/hard'',
``Can we have more applications to Computer Science'', ``Everything's
okay'', and so on. You will not be graded on these comments, but they
will affect how I teach the class, and may make the class more enjoyable for
you.
Class participation: You are expected to actively participate in
class. Many students view learning as a passive act, where the
teacher takes the only active role, and the student simply listens, or
at most takes notes. This view is not advisable in this class. Here,
you will need to take an active role in learning the material. {\em
You} are in charge of your education, and {\em you} should take
responsibility to learn the material as thoroughly as you can. Part
of this involves asking questions in class, even questions that may
sound ``stupid''. A question clearing up a point you do not
understand is, by definition, not stupid. Similarly, when I ask the
class questions, you should try to answer them, even if you're not
sure of the answer. Your best guess is, by definition, not stupid.
There will also be times when the class will discuss a topic and you are
expected to participate in the discussion. Correct answers are irrelevant for
the grading of discussion; all that matters is that you participate in some
meaningful way.
Class participation will be used to decide borderline cases in the
final grade. Remember that since there are 12 grades (counting +'s
and --'s), almost everyone in the class will be a borderline case.
Pre-class preparation: You are expected to read through the section
of the book we are covering before you come to class. If you don't
understand something, write down specific questions you have to ask in class.
Attendance: Attendance is important simply due to the difficulty
of the course. Missing one class may have the effect of your not
being able to follow any of the classes for the rest of the term.
Furthermore, those who do not attend classes will have poor scores on
class participation and this will also affect your grade. In short,
skip class at your peril.
Quizzes: There will be unannounced regular quizzes on the
material to be read. In addition, there will be announced quizzes to
encourage drill on material to be memorized. These will be factored into
the homework score.
Exams: There will be three midterms, and one final. Each midterm
counts for 15% of your grade, the final counts for 25%, and homework
counts for 30%. All midterms and the final will be take-home. Each
midterm is given on Tuesday and due on Friday, at 9 p.m. The final
will be due on Wednesday, Dec 13, at 7:00 p.m. (as described in the
course schedule).
The final exam grade will substitute for your lowest midterm grade if
this is to your advantage. Note that borderline cases will be
resolved by class participation, as noted above.
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Test | Given | Due | | \hline
Midterm 1 | Sep 26 | Sep 29 | |
Midterm 2 | Oct 24 | Oct 27 | |
Midterm 3 | Nov 14 | Nov 17 | |
Final | Dec 5 | Dec 13 | |
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I will hold review sessions before each, at a time that is popular with the
class.
Grading: A grade of C indicates an ability to do homework-like
problems, and memorization of all techniques and definitions. In
order to receive a B, a student must demonstrate a deeper knowledge of
the material, being able to apply the course material to new
circumstances where applicable. An A student must demonstrate this
kind of deep understanding in all of the covered topics, as well as be
able to draw new conclusions from known facts in a logical manner, and
must also demonstrate persistence and dilligence. In the other
direction, a grade of D shows only superficial understanding of the material,
and shows inconsistency to do straightforward problems. An F
grade indicates that the student has severe gaps in even superficial
understanding of the material in the course.
Although this is the philosophy, grading will be done by counting points
received on each problem, as usual. But the difficulty level of the problems
will be arranged in order to achieve the above grading scale.
In addition, students may boost their grade by working on an independent
project that is suitable to the material. Students interested in doing this
should initiate such requests, preferably by suggesting a problem they find
interesting and consulting with me to see if it is appropriate for the course.
Christian attitude: Although not part of the grading for this
course, you are expected to approach this class with a Christian
attitude, being willing to help your fellow classmates to understand
the material outside of class, being willing to be corrected by your
fellow classmates when you see they are right, but firm in your
conviction otherwise, being bold to ask questions without feeling
ashamed of looking foolish, encouraging one another in love, being
patient with those who are asking questions, and preferring a grasp of
the material, which is enduring and becomes part of you, over a grade,
which is transient, external, and shallow. You should diligently
devote the time you spend on this class as to the Lord. As cheating
harms both the cheater and the rest of the class (though in different
ways), you should not cheat, nor should you provide temptations for
others to cheat.
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