Course 18.440, Fall 2014
This course introduces the mathematical framework of probability and random variables. It aims to
provide a rigorous axiomatic development of the theory while, at the same time, building intuition
and problem solving skills.
Some of the topics that we shall cover include: probability spaces; discrete and continuous
random variables; distribution functions; conditional probabilities; joint distributions;
expectations, variances, and higher moments; uniform, binomial, geometric, Poisson, exponential
and Gaussian distributions; Markov, Chebyshev, and Chernov inequalities; the law of large numbers
and the central limit theorem; Markov chains.
Prerequisites: 18.02 or permission of the instructor.
Course: MWF 10-11 AM, 54-100.
TA's: Evgeni Dimitrov (Room E18-401Q) and Aden Forrow (Room E18-301C)
Office hours: Alice Guionnet : Tuesday 2:30--4:30 (E17-310)
Aden Forrow: Thursday 4PM--6PM (E18-358)
Exercises session (or optional recitations): Evgeni Dimitrov : Wednesday 4PM--6PM(66-156)<156>
Any questions regarding grading problems, postponing mid-terms (well in advance) etc should be send to Evgeni Dimitrov (edimitro@mit.edu)
Bibliography:
A First Course in Probability, 9th edition, by Sheldon Ross. Students are welcome to use 6th, 7th, or 8th editions as well. Both hard copies and electronic versions can be found online.
Assignements: Homeworks (20%)[the lowest grade among the 9 homeworks will be forgotten to
compute the averaged homeworks grade], midterms (20% each), final (40%).
There will be about 9 homeworks (tentative due dates: september 12,17,26,
october 3, 24,31,
november 21, 26, december 3)
2 midterms: 50 mn exam during class, on October 17 and November 14 respectively, in Walker.
If you expect to be unable to attend these classes, let Evgeni Dimitrov
know quickly.
Final:Thursday, December 18 from 1:30 to 4:30 PM in JNSN-Track
Problem set 1, due Friday, September 12
Problem set 2, due Monday, September 22
Problem set 3, due Friday, September 26
Problem set 4, due Friday, October 3
Practice with S. Sheffield midterm exam 1 ( Solution )
Practice with S. Sheffield midterm exam 2 ( Solution )
Practice with S. Sheffield midterm exam 3 ( Solution )
Problem set 5, due Friday, October 24
Solution Midterm1, Friday, October 17
Solution Midterm 1, second version, October 23
Problem set 6, due Friday, October 31
Practice Sheffield Problem set, not due
Practice with S. Sheffield midterm 2 exam 1 ( Solution )
Practice with S. Sheffield midterm exam 2 ( Solution )
Practice with S. Sheffield midterm exam 3 ( Solution )
Practice with S. Sheffield midterm exam 4
Problem set 7, due Friday, November 21
Problem set 8, due monday december 1
Solution Midterm 2
Solution Midterm 2 (second)
Problem set 9, due friday december 5
Practice with S. Sheffield final exam 1 ( Solution )
Practice with S. Sheffield final exam 2 ( Solution )
Practice with S. Sheffield final exam 3 ( Solution )
TENTATIVE SCHEDULE
-September 3 and 5: Chapter 1. Counting. Permutations, binomial theorem, multinomial coefficients.
-September 8 to 12: Chapter 2. Probability set-up. Axioms, equal likelihood.
-September 15 and 17: Chapter 3. Conditional probability and Bayes formula.
-September 22 to 26: Chapter 4. 4.1--4.5. Discrete random variables, expectation and variance.
-September 29 to October 3: Chapter 4. 4.6--4.6+9.1. Bernoulli, Poisson, binomial,
and geometric random variables. Poisson process.
-October 6 to 10 : Chapter 5.5.1 and 5.3. Continuous random variables. Uniform random variables.
-October 15: Review, October 17: Midterm.
-October 17 to 22: Chapter 5:5..--5.7. Continuous random variables. Normal random variables, Exponential random variables and more.
-October 24 and 27: Chapter 6: Joint distributions, independent random variables, and conditional distributions.
-October 29 and 31: Chapter 7: 7.1--7.4. Expectation of sums, covariance and correlations.
-November 3 to 7: Chapter 7: 7.5 --7.8: Conditional expectation and moment generating distributions.
-November 12: REVIEW, November 14: Second mid-term exam.
-November 17 to 21: Chapter 8. Weak and strong law of large numbers, central limit theorem.
-November 24 and 26:Strong law of large numbers Chapter 9.Markov Chains.
-December 1 to 3: Chapter 9:Markov chains, entropy.
-December 5 to 10: Review
-December 18: Final.