6.436/15.085  Fundamentals of Probability

Final grades

Dear class,
The scores of the final exam are posted on Stellar. You can pick it up on Monday or Tuesday between 11am and 3pm. Just ask Lynne Dell for it (office 32-D664).
Furthermore, the course grades are also posted online and can be found in your academic record.
It was a pleasure being your TAs this term, and we hope that you have a nice winter break!
Best,

Zied and Martin

Announced on 16 December 2015  4:00  p.m. by Martin Zubeldia

Exam formula sheet

A number of people asked whether the exam sheet will include formulas that may be required, e.g., of common PDFs, etc. The answer is "yes". You do not have to memorize any formulas

Announced on 14 December 2015  1:21  p.m. by John N Tsitsiklis

Review session summary:

Dear class,
Today, during the review session, I covered the following problems:

Final 2011: #1 and #3.
Final 2008: #1, #6 and #7.

Question from student: X_k are iid N(0,1) rvs. Define Y_n = \sum_{n=1}^\infty X_k / 2^k.
a) Show that Y_n converges in distribution to Y~N(0,1/3). (use characteristic functions)
b) Show that Y_n converges to a random variable that has a finite expected value. (use triangle inequality to bound |Y_n| and use corollary 1 of DCT lecture)

If you have questions, please post them on piazza !

Best of luck!
Zied and Martin

Announced on 12 December 2015  1:07  a.m. by Zied Ben Chaouch

Update: Review Session and Office Hours

Dear Class,

1) On Friday, from 2-4pm, Zied will go through problems from past finals and answer your questions.

2) On Monday, from 5-7pm, Martin will hold Office Hours to answer last minute questions.

3) Uncollected problem sets can be picked up on Friday during the review session, or on Monday during Office Hours. You can also drop in the D-666 office to pick them up.

Good luck with your reviews !

Announced on 09 December 2015  5:48  p.m. by Zied Ben Chaouch

Holiday Readings!

The material in the last three lectures is not available in a convenient manner in a single place. Here are some pointers to readings related to topics that we discussed. In general, the book by Williams provides a concise and to-the-point presentation and our development has been fairly close to that book
In addition, the Wikipedia pages for each individual subtopic tend to provide nice overviews.

1. L2 spaces and projection theorem: Wlliams, Sections 6.8-6.11

2. Conditional expectation: Williams, Chapter 9.

3. (Regular) conditional distributions are briefly discussed in Williams, Section 9.9, and more thoroughly in Dudley, Real Analysis and Probability, Section 10.2

4. Radon-Nikodym derivative. See the Wikipedia page
https://en.wikipedia.org/wiki/Radon–Nikodym_theorem
for a statement and the simplest available proof. Other proofs are poossible:

a) A proof using functional-analytic methods (the Riesz representation theorem); see e.g., Dudley, Real Analysis and Probability, Section 5.5.

b) A proof using martingale methods. Here, one approximates the probability model through a sequence of essentially discrete probability models, and takes the limit.

The martingale proof involves heavy use of conditional expectations to establish the Radon-Nikodym theorem. In a converse approach, one can start with the Radon-Nikodym theorem (e.g., proved using analysis tools), and apply it to define conditional expectations, as in Dudlley, Section 10.1, or
https://en.wikipedia.org/wiki/Conditional_expectation

Announced on 08 December 2015  5:42  p.m. by John N Tsitsiklis