18.440 Probability and Random Variables: Spring, 2014
Lectures: MWF 1112, 54100
Office hours: Wednesday 4 to 6, E17312
TA: Dimiter Ostrev
TA office hours: Thursday 35, E17301W
Text: A First Course in Probability, 8th edition, by Sheldon Ross.
(Here's another free
and funtoread book you might enjoy.) I am using the 8th edition,
but students are welcome to use 6th, 7th, or 9th editions as well. Both hard copies and
electronic versions can be found online (e.g., by searching for Sheldon
Ross probability at amazon.com or google.com or ebay.com...)
Assignments: Homeworks (20%), midterm exams (40%), final exam (40%)
Final
exam on Tuesday, May 20 from 1:30 to 4:30 PM on Johnson Track
Stellar course web site
TENTATIVE SCHEDULE

Lecture 1 (February 5): 1.11.3 Permutations and
combinations
(also Pascal's
triangle  as
studied (not invented) by Pascal, see also
correspondence with Fermat).
 Lecture
2 (February 7): 1.41.5 Multinomial coefficients
and more
counting (see
Pascal's pyramid)
Problem Set One, due
February 14

Lecture 3 (February 10): 2.12.2 Sample spaces and set
theory

Lecture 4 (February 12): 2.32.4 Axioms of probability
(see
Paulos' NYT article and a famous hat
problem)

Lecture 5 (February 14): 2.52.7 Probability and equal likelihood
(and a bit more
history )
Problem Set Two, due February 24

Lecture 6 (February 18, Tuesday): 3.13.2
Conditional probabilities

Lecture 7 (February 19): 3.33.5 Bayes' formula and independent
events

Lecture 8 (February 21): 4.14.2 Discrete random variables
Problem Set Three, due February 28

Lecture 9 (February 24): 4.34.4 Expectations of discrete random
variables (and, for nondiscrete setting, examples of nonmeasurable sets,
as in the Vitali construction)

Lecture 10 (February 26): 4.5 Variance

Lecture 11 (February 28): 4.6 Binomial random variables, repeated
trials and the socalled Modern Portfolio Theory.
Problem Set Four, due
March 7

Lecture 12 (March 3): 4.7
Poisson random variables

Lecture 13 (March 5): 9.1 Poisson processes

Lecture 14 (March 7): 4.84.9 More discrete random variables
Practice Midterm Exam
with partial solutions (here is an old
midterm
and 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 , and Fall
2012
midterm with
solutions.

Lecture 15 (March 10): 5.15.2 Continuous random variables

Lecture 16 (March 12): REVIEW

Lecture 17 (March 14):
First Midterm with
solutions
Problem Set Five, due March 21

Lecture 18 (March 17): 5.3 Uniform random variables

Lecture 19 (March 19): 5.4 Normal random variables

Lecture 20 (March 21): 5.5 Exponential
random variables
Problem Set Six, due April 4

Lecture 21 (March 31): 5.65.7 More continuous random variables

Lecture 22 (April 2): 6.16.2 Joint distribution functions

Lecture 23 (April 4): 6.36.5 Sums of independent random
variables
Problem Set Seven, due
April 11

Lecture 24 (April 7): 7.17.2 Expectation of sums

Lecture 25 (April 9): 7.37.4 Covariance and correlation.
REMARK: Everyone knows that
correlation does not imply causation... or do they? Try typing
"study"
and "linked to" into google and page through until you find 100 distinct
headlines roughly of the form "Study links A to B". How many
seem to be real correlations (as opposed to statistical flukes or chance
anomalies or measurement/methodology errors
that might not appear
in a larger,
more careful study)? Do authors the provide the info 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 the most obvious causal and
noncausal explanations?

Lecture 26 (April 11): 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 and
Fall 2012 second midterm with
solutions

Lecture 27 (April 14): 7.77.8 Moment generating distributions

Lecture 28 (April 16): REVIEW

Lecture 29 (April 18):
Second Midterm with
solutions
Problem Set Eight, due
April 25

Lecture 30 (April 23): 8.18.2 Weak law of large numbers

Lecture 31 (April 25): 8.3 Central limit theorem
Problem Set Nine, due May 2

Lecture 32 (April 28): 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 30): 9.2 Markov chains

Lecture 34 (May 2): 9.39.4 Entropy
Problem Set Ten, due May 9
(see this short
martingale note for supplemental reading)

Lecture 35 (May 5): Martingales and the
Optional Stopping Time
Theorem
(see also prediction market plots)

Lecture 36 (May 7): Risk Neutral Probability and BlackScholes
(look up options quotes at the Chicago Board Options Exchange)

Lecture 37 (May 9): REVIEW

Lecture 38 (May 12:) REVIEW

Lecture 39 (May 14): REVIEW
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 .
 May 20:
Final exam on Johnson Track from 1:30 to 4:30 PM... and here is the
actual Spring 2014 Final
with
solutions .