18.600 Probability and Random Variables: Spring 2025

Lectures: MWF 1-2 in 54-100, see also Canvas site

Office hours: Friday 2-4 in 2-249

TA Recitations: Problem solving on Thursdays starting September 11

  Kathy Choi: 10-11 Thursdays in 2-147

  Ivan Motorin: 11-12 Thursdays in 2-147

  Joshua Messing: 12-1 Thursdays in 2-147

  Korina Digalaki: 1-2 Thursdays in 2-147

TA and UA office hours:

  Joshua Messing: 2pm-4pm Wednesdays in 13-5101

  Kathy Choi: 5pm-7pm Wednesdays in 2-135

  Korina Digalaki: 2pm-4pm Thursdays in 1-371

  Yaman Otuzbir: 6pm-8pm Thursdays in 2-135

  Tiffany Wang: 10am-noon Fridays in 2-131

  Ivan Motorin: 7pm-9pm Fridays in 2-131

Text: A First Course in Probability, by Sheldon Ross. I use the 8th edition, but students are welcome to use 6th, 7th, 9th, or 10th editions as well. Both hard copies and electronic versions can be obtained inexpensively online by looking up "first course in probability" via google, amazon, ebay, etc. (Here's another free and fun book.)

Assignments: 10 problem sets (50%), 2 midterm exams (25%), 1 final exam (25%)

Final exam: time and place TBA. Check registrar posting for updates.

Gradebook: managed on Canvas site

Numbering note: Until spring 2015, the course now called 18.600 was called 18.440. It was renamed as part of a departmental effort to make course labels more logical. The current label conveys that 18.600 is a foundational class and a starting point for the 18.6xx series.

Story sheet: This story sheet contains things you really should know by heart. Math fluency requires knowing at least few things by heart: Pythagorean theorem, definition of sine, etc. The red items on the story sheet are things students should know (or be able to quickly derive) by the end of the course: the "basic discrete random variables" by the first midterm, and the "basic continuous random variables" and "moment generating and characteristic function" facts by the second midterm. Try to learn the story that goes with each concept while it is being covered. Some of these items are pretty easy to remember (or deduce from basic principles) once you have the concepts down.

Merged lectures: Here is a printable pdf file containing a preliminary version of all of the lectures for the course. You can print this out and take notes on it during lecture if this is helpful. (I left outline pages in, so there should be room for notes there.) Note that if changes are made to the slides during the semester, they won't necessarily be updated on this document.

Warmup: lecture warm-up song zip aided by local AI

TENTATIVE SCHEDULE

• Lecture 1 (September 3): 1.1-1.3 Permutations and combinations (also Pascal's triangle, history buffs can read a famous correspondence between Pascal and Fermat that helped launch the modern era of probability).
• Lecture 2 (September 5): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid). See Fibonacci's pups. Also Party at Pascal's.

• Problem Set One, due September 12 (students who registered late may submit the first problem set by September 18)

  Lecture 3 (September 8): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (September 10): 2.3-2.4 Axioms of probability (see Paulos' NYT article and conjunction fallacy blog and famous hat problem)
• Lecture 5 (September 12): 2.5-2.7 Probability and equal likelihood (and a bit more history)

  Problem Set Two, due September 18

• Lecture 6 (September 15): 3.1-3.2 Conditional probabilities
• Lecture 7 (September 17): 3.3-3.5 Bayes' formula and independent events

  Problem Set Three, due September 26

• Lecture 8 (September 22): 4.1-4.2 Discrete random variables
• Lecture 9 (September 24): 4.3-4.4 Expectations of discrete random variables (and, for non-discrete setting, examples of non-measurable sets, as in the Vitali construction)
• Lecture 10 (September 26): 4.5 Variance

  Problem Set Four, due October 3

• Lecture 11 (September 29): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.
• Lecture 12 (October 1): 4.7 Poisson random variables
• Lecture 13 (October 3): 9.1 Poisson processes and 5.1-5.3 Continuous random variables

  Spring 2011 midterm on Chapters 1-4 (plus 9.1) with solutions. Fall 2011 midterm on Chapters 1-4 (plus 5.1-5.4 and 9.1) with solutions. Fall 2012 midterm with solutions. Spring 2014 Midterm with solutions. Spring 2016 midterm with solutions. Spring 2017 midterm with solutions. Spring 2018 midterm with solutions. Spring 2019 midterm with solutions. First Fall 2019 midterm with solutions. First Spring 2021 midterm with solutions. First Spring 2022 midterm with solutions. Fall 2023 Midterm 1 with solutions. Spring 2025 Midterm 1 with solutions.

• Lecture 14 (October 6): 4.8-4.9 More discrete random variables
• Lecture 15 (October 8): REVIEW
• Lecture 16 (October 10): Fall 2025 Midterm I

  Problem Set Five. Due October 17

• Lecture 17 (October 15): Uniform random variables
• Lecture 18 (October 17): 5.4 Normal random variables

  Problem Set Six, due October 24

• Lecture 19 (October 20): 5.5 Exponential random variables
• Lecture 20 (October 22): 5.6-5.7 More continuous random variables
• Lecture 21 (October 24): 6.1-6.2 Joint distribution functions

  Problem Set Seven, due October 31

• Lecture 22 (October 27): 6.3-6.5 Sums of independent random variables
• Lecture 23 (October 29): 7.1-7.2 Expectation of sums
• Lecture 24 (October 31): 7.3-7.4 Correlation and causation

  Problem Set Eight, due November 7

• Lecture 25 (November 3): 7.5-7.6 Conditional expectation
• Lecture 26 (November 5): 7.7-7.8 Moment generating functions
• Lecture 27 (November 7): 8.1-8.2 Weak law of large numbers

  Spring 2011 second midterm on 1-7 (plus 9.1) with solutions. Fall 2011 second midterm with solutions. Fall 2012 second midterm with solutions. Spring 2014 second midterm with solutions. Spring 2016 second midterm with solutions. Spring 2017 second midterm with solutions. Spring 2018 second midterm with solutions. Second Spring 2019 Midterm with solutions. Second Fall 2019 Midterm Two with solutions. Second Spring 2021 Midterm with solutions. Second Spring 2022 Midterm with solutions. Second Fall 2023 Midterm with solutions. Second Spring 2025 Midterm with solutions.

• Lecture 28 (November 12): REVIEW
• Lecture 39 (November 14): Second Spring 2025 Midterm

  Problem Set Nine, due November 21

• Lecture 30 (November 17): 8.3 Central limit theorem
• Lecture 31 (November 19): 8.4-8.5 Strong law of large numbers (see also the truncation-based proof on Terry Tao's blog and the characteristic function proof of the weak law) and Jensen's inequality.
• Lecture 32 (November 21): 9.2 Markov chains

  Problem Set Ten, due December 5 (see this short martingale note for supplemental reading)

• Lecture 33 (November 24): 9.3-9.4 Entropy
• Lecture 34 (November 26): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)

  Lecture Notes for 34 to 36 which follow the outline of the lecture slides and cover martingales, risk neutral probability, and Black-Scholes option pricing (topics that do not appear in the textbook, but that are part of this course)

• Lecture 35 (December 1): Martingales and Risk Neutral Probability (look up options quotes at the Chicago Board Options Exchange)
• Lecture 36 (December 3): Call functions and Black-Scholes
• Lecture 37 (December 5): Review

• Lecture 38 (December 8): Review
• Lecture 39 (December 10): Review

  Practice Final Problems (covering only later portion of the course) with partial solutions. Spring 2011 final with solutions. Fall 2012 final with solutions. Spring 2014 final with solutions. Spring 2016 final with solutions. Spring 2017 final with solutions. Spring 2018 final with solutions. Spring 2019 Final Exam with solutions. Fall 2019 Final Exam with solutions. Spring 2021 Final Exam with solutions. Spring 2022 Final Exam with solutions. Fall 2023 Final Exam with solutions Spring 2025 Final Exam with solutions .