18.600 Probability and Random Variables: Spring 2025

Lectures: MWF 2-3 in 34-101, see also Canvas site

Office hours: Friday 3-5 in 2-249

TA Recitations: Problem solving on Thursdays starting February 13

  Korina Digalaki: Thursday 10am-11am in 32-144

  Mikayel Mkrtchyan: Thursday noon-1pm in 2-190

  Jason Yang: Thursday 3pm-4pm in 2-190

  Yuchong Pan: Thursday 4pm-5pm in 4-163

TA and UA office hours:

  Jason Yang: Tuesdays 2pm-4pm in 38-166

  Korina Digalaki: Wednesdays 3pm-5pm in 2-132

  Mikayel Mkrtchyan: Thursdays 1pm-3pm in 2-242

  Yuchong Pan: Fridays 12pm-2pm in 2-242

  Joshua Lee: Thursdays 5pm to 7pm in 2-151

  Enrique Rivera Ferraiuoli: Fridays 7pm-9pm 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: TBD 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.

TENTATIVE SCHEDULE

• Lecture 1 (February 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 (February 5): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid). See Fibonacci's pups. Also Party at Pascal's.
• Lecture 3 (February 7): 2.1-2.2 Sample spaces and set theory

  Problem Set One, due February 14 (students who registered late may submit the first problem set by February 21)

• Lecture 4 (February 10): 2.3-2.4 Axioms of probability (see Paulos' NYT article and conjunction fallacy blog and famous hat problem)
• Lecture 5 (February 12): 2.5-2.7 Probability and equal likelihood (and a bit more history)
• Lecture 6 (February 14): 3.1-3.2 Conditional probabilities

  Problem Set Two, due February 21

• Lecture 7 (February 18): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (February 19): 4.1-4.2 Discrete random variables
• Lecture 9 (February 21): 4.3-4.4 Expectations of discrete random variables (and, for non-discrete setting, examples of non-measurable sets, as in the Vitali construction)

  Problem Set Three, due February 28

• Lecture 10 (February 24): 4.5 Variance
• Lecture 11 (February 26): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.
• Lecture 12 (February 28): 4.7 Poisson 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.

• Lecture 13 (March 3): 9.1 Poisson processes and 5.1-5.3 Continuous random variables
• Lecture 14 (March 5): REVIEW
• Lecture 15 (March 7): First Spring 2025 Midterm

  Problem Set Four, due March 14

• Lecture 16 (March 10): 4.8-4.9 More discrete random variables
• Lecture 17 (March 12): Uniform random variables
• Lecture 18 (March 14): 5.4 Normal random variables

  Problem Set Five. Due March 21

• Lecture 19 (March 17): 5.5 Exponential random variables
• Lecture 20 (March 19): 5.6-5.7 More continuous random variables
• Lecture 21 (March 21): 6.1-6.2 Joint distribution functions

  Problem Set Six, due April 4

• Lecture 22 (March 31): 6.3-6.5 Sums of independent random variables
• Lecture 23 (April 2): 7.1-7.2 Expectation of sums
• Lecture 24 (April 4): 7.3-7.4 Correlation and causation

  Problem Set Seven, due April 11

• Lecture 25 (April 7): 7.5-7.6 Conditional expectation
• Lecture 26 (April 9): 7.7-7.8 Moment generating functions
• Lecture 27 (April 11): 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.

• Lecture 28 (April 14): 8.3 Central limit theorem
• Lecture 29 (April 16): REVIEW
• Lecture 39 (April 18): Second Spring 2025 Midterm

  Problem Set Eight, due April 25

• Lecture 31 (April 23): 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 (April 25): 9.2 Markov chains

  Problem Set Nine, due May 2

• Lecture 33 (April 28): 9.3-9.4 Entropy
• 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 34 (April 39): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)
• Lecture 35 (May 2): Martingales and Risk Neutral Probability (look up options quotes at the Chicago Board Options Exchange)

  Problem Set Ten, due May 9 (see this short martingale note for supplemental reading)

• Lecture 36 (May 5): Call functions and Black-Scholes
• Lecture 37 (May 7): Review
• Lecture 38 (May 9): Review

• Lecture 39 (May 12): 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