18.600 Probability and Random Variables: Spring 2016
Lectures: MWF 1011 in 54100
Office hours: Wednesday 3:00 to 5:00 in 2249
TAs: Cesar Cuenca and Hong Wang
Hong Wang's office hours:: Monday 4:30 to 6:30 in
2231
Cesar Cuenca's office hours: Thursday 5:00 to 7:00 in
2231
Text: A First Course in Probability, by Sheldon Ross.
I use the 8th edition,
but students are welcome to use 6th, 7th, or 9th editions as well. Both hard copies and
electronic versions can be obtained inexpensively online by looking up "first
course in probability" at Google, Amazon, or ebay. (Here's another free
and fun book.)
Assignments: 10 problem sets (20%), 2 midterm exams (40%), 1 final exam (40%)
Final exam: Johnson Track Tuesday, May 17, 912.
Check registrar posting for updates.
Gradebook: managed on
Stellar course web site
Numbering note: Until spring 2015, the course now called 18.600 was called 18.440. It was renamed last year as part of a departmental effort to make course labels more
logical. The current label conveys that 18.600 is a foundational class and a starting
point for the 18.6xx series.
TENTATIVE SCHEDULE

Lecture 1 (February 3): 1.11.3 Permutations and
combinations
(also Pascal's
triangle  as
studied (not invented) by Pascal, see also
correspondence with Fermat).
 Lecture
2 (February 5): 1.41.5 Multinomial coefficients
and more
counting (see
Pascal's pyramid)
Problem Set One, due
February 12 (students who have registered late may submit the first problem set on
February 19)

Lecture 3 (February 8): 2.12.2 Sample spaces and set
theory

Lecture 4 (February 10): 2.32.4 Axioms of probability
(see
Paulos' NYT article and a famous hat
problem)

Lecture 5 (February 12): 2.52.7 Probability and equal likelihood
(and a bit more
history )
Problem Set Two, due
February 19

Lecture 6 (February 16, Tuesday): 3.13.2
Conditional probabilities

Lecture 7 (February 17): 3.33.5 Bayes' formula and independent
events

Lecture 8 (February 19): 4.14.2 Discrete random variables
Problem Set Three, due
February 26

Lecture 9 (February 22): 4.34.4 Expectations of discrete random
variables (and, for nondiscrete setting, examples of nonmeasurable sets,
as in the Vitali construction)

Lecture 10 (February 24): 4.5 Variance

Lecture 11 (February 26): 4.6 Binomial random variables, repeated
trials and the socalled Modern Portfolio Theory.
Practice Midterm Exam
with partial solutions.
2009 Midterm One With
Solutions.
Spring
2011 midterm exam
on Chapters 14 (plus 9.1) with
solutions.
Fall
2011 midterm exam
on Chapters 14 (plus 5.15.4
and 9.1) with
solutions . Fall
2012
midterm with
solutions.
Spring 2014 Midterm with
solutions

Lecture 12 (February 29): 4.7
Poisson random variables

Lecture 13 (March 2): REVIEW

Lecture 14 (March 4):
First midterm with
solutions
Problem Set Four, due
March 11

Lecture 15 (March 7): 9.1 Poisson processes

Lecture 16 (March 9): 4.84.9 More discrete random variables

Lecture 17 (March 11): 5.15.2 Continuous random variables
Problem Set Five, due March
18

Lecture 18 (March 14): 5.3 Uniform random variables

Lecture 19 (March 16): 5.4 Normal random variables

Lecture 20 (March 18): 5.5 Exponential
random variables
Problem Set Six, due April 1

Lecture 21 (March 28): 5.65.7 More continuous random variables

Lecture 22 (March 30): 6.16.2 Joint distribution functions

Lecture 23 (April 1): 6.36.5 Sums of independent random
variables
Problem Set Seven, due
April 8

Lecture 24 (April 4): 7.17.2 Expectation of sums

Lecture 25 (April 6): 7.37.4 Covariance and correlation. (Fun
weekend activity:
see how
humans think about correlation and
causation by typing
"study"
and "linked to" into google and paging through until you find 100 distinct
"Study links A to B" headlines. How many do you think are real
correlations (as opposed to statistical flukes or chance
anomalies or measurement/methodology errors
that might not appear
in a larger,
more careful study)? In how many cases do the authors provide what you
need
to assess (1) the strength of evidence for
correlation existence (2) magnitude of the reported correlation (3)
what is known about the plausibility of obvious causal and
noncausal explanations?)

Lecture 26 (April 8): 7.57.6 Conditional expectation
Practice Midterm Exam Two
with
partial solutions and 2009 Midterm Two
with solutions
.
Spring 2011 Second midterm exam on 17 (plus 9.1) with
solutions.
Fall 2011 second midterm with solutions.
Fall 2012 second midterm with
solutions.
Spring 2014 second Midterm with
solutions. There will be no formula sheet on the exam, but
here is a
story sheet which you can use to prepare in advance (but not during
the exam itself; instead of memorizing, try to understand the stories well
enough that remembering the formulas is automatic).

Lecture 27 (April 11): 7.77.8 Moment generating distributions

Lecture 28 (April 13): REVIEW

Lecture 29 (April 15):
Second
midterm with
solutions.
Problem Set Eight, due April 22

Lecture 30 (April 20): 8.18.2 Weak law of large numbers

Lecture 31 (April 22): 8.3 Central limit theorem
Problem Set Nine, due April
29

Lecture 32 (April 25): 8.48.5 Strong law of large numbers (see
also
the truncationbased proof on Terry Tao's blog and the characteristic
function proof of the weak law) and Jensen's inequality.

Lecture 33 (April 27): 9.2 Markov chains

Lecture 34 (April 29): 9.39.4 Entropy
Problem Set Ten, due May
6
(see this short
martingale note for supplemental reading)

Lecture 35 (May 2): Martingales and the
Optional Stopping Time
Theorem
(see also prediction market plots)

Lecture 36 (May 4): Risk Neutral Probability and BlackScholes
(look up options quotes at the Chicago Board Options Exchange)

Lecture 37 (May 6): REVIEW

Lecture 38 (May 9:) REVIEW

Lecture 39 (May 11): REVIEW
Problem outtakes (which for various reasons did not make it into
a problem set, but which you can browse if you are curious)
Practice Final Problems
(covering
only later portion of the course) with
partial solutions and
Spring 2011
Final
with
solutions and
Fall 2012
Final
with
solutions .
Spring 2014 Final
with
solutions
.

Final
exam (with
solutions): Johnson Track, Tuesday, May 17, 912.
Check registrar posting for updates.