18.440 Probability and Random Variables
MIT Department of Mathematics, Spring 2007
February 7, 2007
Information Sheet
Course Website: http://math.mit.edu/18.440
Lecturer
Shan-Yuan Ho
Office: 2-346
Tel: 617-324-2614
E-mail: hoho@math.mit.edu
Office hours: Tues. 3-4 & Wed. 1-2 & by appointment.
Teaching Assistant
Linan Chen
lnchen@math.mit.edu
Office: 2-342
Tel: 617-253-7578
Teaching Assistant
Fang Wang
fang@math.mit.edu
Office: 2-251
617-253-7566
Teaching Assistant
Zhenqi He
zhenqi@math.mit.edu
Office: 2-088
617-253-1194
TA Office Hours: held in Room 2-349 on
Mon. 4:30-5:30 & Tues 4:30-6:30 & Tues 7-8 & by appointment.
Time and Place
12-1 PM on Monday, Wednesday, Friday in Room 2-190.
Course Description
This course presents the mathematical framework of probability theory and is concerned
with the nature, formulation, and analysis of probabilistic situations. It is a calculusbased course with emphasis on developing probability concepts, intuitive interpretations,
and problem solving skills. Topics covered include probability spaces, random variables,
distribution functions, conditional probability, Bayes’ rule, binomial, geometric, hypergeometric, Poisson distributions, uniform, exponential, Gaussian/normal, gamma and beta
distributions, joint distributions, Chebyshev inequality, Law of Large Numbers, and Central Limit Theorem.
Credits
This is a twelve credit subject, (3-0-9).
Prerequisites
1
18.02 or equivalent. No previous experience with probability assumed.
Text
Sheldon Ross, A First Course in Probability, 7th ed., Prentice Hall, 2005.
ISBN: 0131856626.
Problem Sets
There will be 11 problem sets, corresponding to a weekly schedule, though the final
problem set will not be collected. You are expected to do all the assigned problems, and
we will assume that in making up the exams and final. We encourage you to cooperate
with each other in doing the problem sets. The problem sets are vehicles for learning,
and whatever maximizes learning for you is desirable. This usually includes discussion,
teaching of others, and learning from others. It is most beneficial to discuss homework
problems with classmates and friends (as well as the TA or the lecturer) AFTER you’ve
made an effort to think about them by yourself for a while. It is worthwhile for you to
develop the habit and ability to discuss mathematics with others, since discussions can
be a valuable way to gain insight and familiarity.
Collaboration does not mean copying someone else’s work in what you hand in. Write
up your problem sets independently. Merely copying papers from others circumvents the
learning process and you should avoid it. A good way to proceed is to work out the idea of
a solution with classmates, but then write it up alone, in your own words, without relying
on detailed notes (you should have absorbed the key ideas and internalized them). If your
final write-up looks too much like your collaborators’ write-up, you are probaby leaning
on the group too much in the writing phase and thereby missing out on the valuable
experience of writing up something on your own. Employers of scientists and engineers
regard communication skills as having as much importance as mathematical skills, so it
literally pays to develop these skills.
Include on each problem that you hand in a list of the people you worked with on that
problem or other sources your consulted (outside the textbook and lectures). This information will in no way be used in grading. It is intended to help you get into the life-long
habit of citing sources and avoiding plagiarism. (Remember, plagiarism isn’t copying; it’s
copying without acknowledgment.)
Problem sets must be handed in by 4PM at the UMO (Undergraduate Math
Office, Room 2-108) on the due date. Problem set solutions will be posted shortly
afterward on the course website. Consequently, it is difficult and unfair to seriously
evaluate late problem sets, so no late problems sets will be accepted. If a note from
the Dean’s office or Student Support Services is provided, you will be excused from the
appropriate problem sets.
There are usually multiple ways to approach probability problems. An agreement with
the correct answer does not necessarily mean that the approach is correct. It is important
that you understand where your thought processes might have been flawed. Please consult
2
the teaching staff if you need help. You are welcome to flag topics or problems of confusion
to you in the problem sets; this will not lower your grade.
Exams
There will be two quizzes given during the semester in the regular lecture location and
time. A final exam will be given during the scheduled final exam period. The quizzes
and final will be closed book, but you may bring one double sided 8.5” by 11” sheet of
notes to Quiz 1, two to Quiz 2, and three double sided 8.5” by 11” sheets of notes to the
Final Exam. Most people find that the process of preparing such notes helps them much
more than their use. Since your grade on the exams will reflect the degree to which you
demonstrate an understanding of the material, it pays to be honest; it’s better to say,
”This answer does not make any sense or looks wrong for such-and-such a reason... ,”
rather than to bluff.
The quiz dates are on Wednesday, March 14 and Wednesday, April 18. The final exam
and the conflict exam will be scheduled by the registrar for 3 hours during the third week
of the term. We will attempt to make each quiz and the final exam a test of understanding
rather than of speed-writing.
Course Grade
The final grade in the course is based upon our best assessment of your understanding
of the material. This assessment is based on four noisy measurements: the problem sets,
Quiz 1, Quiz 2, and the Final Exam.
The different measurements have different noise levels, and the final grade will be a
weighted average, roughly according to the following rule:
Problem Sets:
Quiz 1:
Quiz 2:
Final Exam:
10%
25%
25%
40%
Reference Texts
(1) D.P. Bertsekas and J.N. Tsitsiklis, Introduction to Probability, Athena Scientific,
2002.
(2) R.D. Yates and D.J. Goodman, Probability and Stochastic Processes, 2d ed., Wiley,
2005.
(3) W. Feller, An Introduction to Probability Theory and Its Applications, Volume 1,
Wiley, 1950.
3
Problem Sets
Out
Due
Day Date
1
Wed 2/7
Fri
2/9
Mon 2/12
2
1
Wed 2/14
Fri
2/16
Tues 2/20
3
2
Wed 2/21
Fri
2/23
Mon 2/26
4
3
Wed 2/28
Fri
3/2
Mon 3/5
5
4
Wed 3/7
Fri
3/9
Mon 3/12
Wed 3/14
Fri
3/16
Mon 3/19
6
5
Wed 3/21
Fri
3/23
3/26–3/30
Mon 4/2
7
6
Wed 4/4
Fri
4/6
Mon 4/9
8
7
Wed 4/11
Fri
4/13
4/16–4/17
Wed 4/18
Fri
4/20
Mon 4/23
9
8
Wed 4/25
Fri
4/27
Mon 4/30
10
9
Wed 5/2
Fri
5/4
Mon 5/7
11
10
Wed 5/9
Fri
5/11
Mon 5/14
Wed 5/16
5/21–5/25
#
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5
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8
9
10
11
12
13
14
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18
19
20
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23
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25
26
27
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29
30
31
32
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Lectures
Topic (tentative - subject to change)
Introduction
Combinatorics
Sample Space, Events
Axioms of Probability
Equally likely Outcomes
Conditional Probability
Bayes’ Rule
Independent events
Random Variables
Expected Value, Variance
Distributions: Bernouli and Binomial
Distributions: Poisson
Distributions: Geometric and others
Cumulative Distribution Function
Review and Practice Test
Quiz 1
Conditional Distributions
Continuous r.v., density functions
Distributions: Uniform
Distributions: Gaussian/normal
Spring Break
Distributions: Exponential
Derived Distributions
Joint Distribution Functions
Sums of Independent r.v.
Conditional Distributions
Review and Practice Test
Patriot’s Vacation
Quiz 2
Functions of r.v.
Random sums of random variables
Conditional Expectation, Conditional Variance
Covariance and Correlations
Moment Generating Functions
Markov and Chebyshev’s Inequality
Central Limit Theorem
Law of Large Numbers
Bounding Error Probabilities
Poisson Process
Markov Chains
Random Walks
FINAL EXAM PERIOD
4
Reading
Chap 1
Chap 1
Chap 2
Chap 2
Chap 2
Chap 3
Chap 3
Chap 3
Chap 4
Chap 4
Chap 4
Chap 4
Chap 4
Chap 4
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