21-425: Probability and Martingales

Fall 2026

Course Information

Discussion Board / Mailing List

Office Hours

Time Place Person
Mondays 2:15pm – 3:15pm WEH 8115 Gautam Iyer
Fridays 12:00pm – 1:00pm WEH 6205 Carlos Matos

Homework, Notes, etc.

Exam Dates

  • Midterm 1: Wed, Sep 16 (4th week), closed book, in class.
  • Midterm 2: Wed Oct 21st (8th week), closed book, in class.
  • Midterm 3: Wed Nov 18th (12th week), closed book, in class.
  • Final: Comprehensive, closed book, in class. The time of the final will be announced by the registrar here. Be aware of their schedule before making your travel plans.

More information on the exam format is here.

Grading

Your scores on exams and homework will each be converted to common scale using cutoffs announced after each exam. Your exam average will be computed as the maximum of your converted midterm / final scores with the following weights:

  • 20% each midterm, 40% final.
  • 20% best two midterms, 60% final.

Your overall grade will be the minimum of your exam average, and your homework average. (See the homework policies and exam format for how significantly homework influences your grade.)

Course description.

This course serves as a rigorous introduction to probability and is an accelerated version of 21-325 (Probability). Probability is a fundamental tool that has found applications in most modern scientific fields. It will introduce the fundamentals (probability spaces, random variables, distributions) and then study independence, conditioning, the law of large numbers, the central limit theorem, martingales, stopping times, optional sampling, and optimal stopping problems. No previous familiarity with probability is required.

Tentative Syllabus

  • Sample spaces, random variables, distributions.
  • Conditional probability, independence
  • Expectation, variance, moment generating functions
  • Random walks, sums of independent random variables, convolutions.
  • Borel Cantelli, Law of Large numbers
  • Normal distributions, characteristic functions, central limit theorem.
  • Conditional expectation, Martingales
  • Stopping times, Doob’s optional sampling
  • Supermartingale envelopes and optimal stopping
  • Recurrence of random walks (time permitting)

Learning Objectives

  1. Understand the foundations of Probability.
  2. Develop familiarity with advanced topics such conditional expectations and martingales
  3. Develop logical thinking / problem solving skills.

Pre-requisites

  • Familiarity with writing proofs (A in 21127 or 15151 or 21128).
  • Basic calculus (A in 21259 or 21266, or at least a B in 21268 or 21269).

References

The material covered in this course is standard and can be found in many good references. As a result my notes above are extremely brief, and will only have statements of definitions and theorems. Proofs and intuition will be done in class; if you miss class I suggest reading this from one of many standard references. The specific choice of topics we cover may not be done similarly, with the same notation, or in the same order in the references below.

  • W. Feller, Introduction to Probability Theory and Its Applications, Volume I
  • G. Grimmett, D. Welsh, Probability: an introduction
  • R. Durrett, Probability: Theory and Examples
  • D. Williams Probability with Martingales

Class Policies

Lectures

  • If you must sleep, don’t snore!
  • Be courteous when you use mobile devices

Homework

The primary purpose of homework is to help you understand the material!

Homework policy

  • You may collaborate, use AI/online resources, etc. freely,
  • Homework will be graded on an extremely generous curve chosen so that anyone who is honestly works on the homework will get an A on the homework. (As a result, by the grading policy and your grade will most likely be entirely determined by your performance on exams.)
  • A significant portion of your exams will consist of homework problems, as described here. Your solutions to exam questions will be scrutinized more carefully, and you will be held to a higher, and more rigorous standard, on exams.
  • No homework solutions will be posted! This is intentional. The most important skill in math is finding solutions, and recognizing correct solutions; to dis-incentivise rote memorization of solutions the night before exams, I will not post any.

The above policy was chosen because some version of most problems we will encounter in this class has already appeared in several standard references. These have been thoroughly indexed by most AI models and discussed extensively in online forums. You may get very good solutions from AI models (or the internet) easily and quickly. These are valuable resources – if used properly. You may use it freely, but there is very little grade incentive for you to blindly copy.

A significant portion of your closed book, NO AI / Internet / collaboration, exams will consist of homework problems. A good, solid, understanding of the fundamental ideas behind the solution will serve you well on exams. Spending time thinking about these problems, and trying to solve it yourself, or understanding ideas / solutions you found will also help. However, if you blindly copy solutions, you will likely have a very hard time before exams.

Recommendations

  • Start the homework early. You learn a lot more by thinking about a problem for longer. (Also, most students won’t be able to do the homework in one evening.)
  • Go over homework problems you didn’t get correct, and understand correct solutions thoroughly. (While solutions won’t be posted, you can get them from us in person, or your favorite friend / internet source / AI model.)
  • Collaboration, using the internet, AI, etc. is 100% legal. If you get a solution from someone else, just be sure you understand it completely and thoroughly. (It may also help if you understood how your source arrived at the solution.)

Logistics.

  • All homework must be scanned and turned in via Gradescope. (Everyone who was registered for this class on day 1 was automatically added to Gradescope. If you joined later, use this invite code to add yourself.)
  • Please take good quality scans; homework that’s too hard to read won’t be graded. I recommend using a good scanning app that adjusts the contrast of your images for readability. (I’ve had good luck with Adobe Scan, and Google Drive.)
  • Homework solutions will not be posted!

Late Homework Policy

  • You may turn in up to four homework assignments up to 24 hours late without penalty.
  • I will drop the two lowest two homework scores from your grade.
  • For emergencies or special circumstances, you may drop one additional homework score from your grade.
  • If you have a circumstances that may warrant allowances beyond the above (e.g. health issues that puts you out for a few weeks), then please contact your academic advisor first, and have them reach out to me.

Exams

Content

  • All midterms, and half the final will consist of (randomly chosen) homework questions, or results done in class. If the chosen question is too long for the exam, I will simplify it so that it is either a special case, or a step in the proof.
  • Half the final will be Ph.D. Thesis interesting problems which you may or may not have seen.

Logistics

  • The final time will be announced by the registrar here. Be aware of their schedule before making your travel plans.
  • No calculators, computational aids, or internet enabled devices are allowed for closed book exams.
  • No makeup midterms will be given. The grading policy allows you to miss one midterm without affecting your grade. If you have circumstances that warrant allowances beyond the above, then please contact your academic advisor first and have them reach out to me.

Academic Integrity

  • All students are expected to follow the academic integrity standards outlined here.
  • There will be zero tolerance for academic integrity violations, and any violation will result in an automatic R. Examples of academic integrity violations include (but are not limited to):
    • Receiving assistance from another person during an exam.
    • Providing assistance to another person taking an exam.
  • All academic integrity violations will further be reported to the university, and the university may chose to impose an additional penalty.

Accommodations for Students with Disabilities

If you have a disability and have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them at access@andrew.cmu.edu.

Note: If due to your course schedule your alternative testing scheduled exam is does not overlap with the time of the regularly scheduled exam, you must contact me two weeks in advance to allow me to make an alternate exam for you. If I’m not notified in a timely manner you will not get credit for an exam taken at a time that does not overlap with the regularly scheduled time.

Student Wellness

As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus here. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

Faculty Course Evaluations

At the end of the semester, you will be asked to fill out faculty course evaluations. Please fill these in promptly, I value your feedback. As incentive, if over 75% of you have filled out evaluations on the last day of class, then I will release your grades as soon as they are available. If not, I will release your grades at the very end of the grading period.