Rice University
This course is the continuation of STAT 581.
The course sequence covers
the measure-theoretic foundations of probability.
Open to qualified
undergraduates.
Test 2 due Wednesday, April 25 (30%) [pdf]
Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020
Office Hours: Monday 3-4 pm
or by appointment
- Sidney Resnick "A Probability Path"
The course will closely follow this book; it is available at the campus bookstore.
20% Homework
20% Quiz (mid Februay; on Convergence Concepts)
30% First EXAM (end March: on sums of independent r.v. and basics on convergence in distribution)
30% Second EXAM (April: on Advanced material on Convergence in Distribution and on Martingales)
Details on tested material: [Knowledge Milestones]
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Classes / Reading
Class contents and reading assignment will be posted here.
Check the schedule of 2006 for an idea on the course contents and progression.
Topics covered reading (Resnick) January 8 no class January 10 Orientation, Review Stat 581: proba spaces, expectation January 12 Review Stat 581: convergence properties of expectations, Borel-Cantelli January 16 No Class (Marting Luther King) January 17 cancelled due to severe weather January 19 I VI. Convergence Concepts
Almost sure, in probability; examplesSection 6.1 (p167) January 19 II (Make-up for January 17) a.s. implies in probability, Cauchy criterium i.p. p169-172, January 22 Cauchy i.P. implies a subsequence converges a.s. p171-174 January 24 Subsequences ("a.s." and "i.P."), 1st Continuity thm, Lebesgue DCT p174-178, p175 January 26 [9-10] Convergence in Lp, relation to "a.s." and "i.P." Section 6.5, p180-182 January 26 [10-11] Uniform integrability Section 6.5, p182-184 January 29 Uniform integrability and moment conditions Section 6.5, p184-186 January 31 Conv L1 <=> conv i.p. & u.i. p190-194 February 2, [9-10] Absolute Continuity & bounded 1st moments <=> u.i. February 2, [10-11] Conv Lp <=> conv i.P. & |X_n|^p is u.i. Inequalities. p186-189 February 5 VII. LLN. Tail equivalence and L2 version of WLLN p203-204 February 7 simple sufficient and the exact equivalent conditions of the WLLN p 205-208 February 9 Sums of indep r.v.: Levy's theorem p209-213 February 12, [9:15] Quiz on chapter 6 (Convergence concepts) February 12, [10-11] Skorohod's inequality, Kolmogorov's convergence criterium February 14 Kronecker's lemma, Records 214-216 February 16 [9:30 - 10:50] Kolmogorov's SLLN 219-222 February 19 Kolmogorov's three series theorem 226-230 February 21 VIII. Convergence in Distribution. Basics, Dense sets 247-248, 248-251 February 23 Vague and weak convergence, relation to convergence in probability 248-251 February 26
Geometric and exponential distr
Scheffe's lemma and convergence in total variation
252-255
Feb 28 Skorohod's theorem
258-260
March 2 [9:30-11]
Continuous mapping, Delta Method
261-263
March 5
Spring Break
March 7
Spring Break
March 9
Spring Break
March 12
no class, made up for on Feb 16 and March 2
March 14
Portmanteau
263-268 March 16
Slutsky's theorem
268-271
March 19
Convergence to Types
274-279
March 21
Extreme Value Distributions
274-279
March 23
Extreme Value Distr. (proof)
Convolution
March 26
IX. Characteristic Function
Simple Properties
293-297
March 28
Expansions of char fct, char fct of the Normal distribution
297-301
March 30
Uniqueness of characteristic functions
302-305
April 2
Continuity of char fct and CLT
312-314
April 4
X. Conditional Expectation: Basics
339-342
April 6
Recess
April 9
Conditional Expectation: simple properties
344-347
April 11
Advanced properties
348-349 + standard refs
April 13
More on martingales: up-crossing and convergence of positive martingales
standard refs
April 16
Hand out exam 2, due last day of class
April 16
Conditional Exp for continuous r.v.
April 18
L2-martingale convergence
April 20
Review
April 23
No Class, made up for on 1/26
April 25
no class, made up for on 2/2
last day of class, exam 2 due
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Homework
(tex-source and solutions restricted
to Rice University)
Homework sheet | Due date (in class) | Solutions |
Problem Set 1 [pdf] [ps] [tex] | Jan 31, 2007 | posted Feb 5 [pdf] [ps] [tex] |
Problem Set 2 [pdf] [ps] [tex] | Feb 7, 2007 |
posted Feb 7 [pdf] [ps] [tex] |
Problem Set 3 [pdf] [ps] [tex] | Feb 19, 2007 |
posted Feb 19 [pdf] [ps] [tex] |
Problem Set 4 [pdf] [ps] [tex] | Feb 28, 2007 |
posted March 1 [pdf] [ps] [tex] |
Practice Exam Set 5 [pdf] [ps] [tex] | March 16, 2007 |
posted March 6 [pdf] [ps] [tex] |
Problem Set 6 [pdf] [ps] [tex] | March 28 |
posted April 4 [pdf] [ps] [tex] |
Problem Set 7 [pdf] [ps] [tex] | April 4 |
posted April 11 [pdf] [ps] [tex] |
Problem Set 8 Practice Test 2[pdf] [ps] [tex] | due April 11 |
Solution 8 posted April 15 [pdf] [ps] [tex] |
Homework is due at the beginning of class on the due date. After the due date, but before solutions are handed out, homework can be turned in for 50% credit. In this case, please slip your homework under the instructors's office door, or bring it to class. After solutions are handed out, 0% credit will be issued. You are encouraged to work in groups for homeworks but you will hand in your own solution which you are expected to understand.
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Quiz |
February 12, in class 9:15 | lecture notes allowed |
Test 1 (30%) [pdf] | Due: March 23 |
Take home, 150 minutes, (open notes) |
Test 2 (30%) [pdf] | Due: last day of class |
Take home, 3 hours (open one book +lecture notes) |
Knowledge Milestones aquired in this
course