PREVIOUS YEAR SITE: 18.600 Probability and Random Variables, Spring 2018

Lectures: MWF 2-3 in 10-250

Office hours: Wednesday 3:00 to 5:00 in 2-249

TAs: Cesar Cuenca and Evgeni Dimitrov

Recitations: Both TAs will offer optional problem solving recitations and officer hours each Thursday (starting February 15).

Cesar Cuenca's recitation: Thursdays at 12:00 in 32-155

Cesar Cuenca's office hours: Thursdays 2:30 to 4:00 in 2-139

Evgeni Dimitrov's recitation: Thursdays at 4:00 in 4-163

Evgeni Dimitrov's office hours: Thursdays 5:00 to 7:30 in 4-153

Text: A First Course in Probability, by Sheldon Ross. I use the 8th edition, but students are welcome to use 6th, 7th, or 9th 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 (20%), 2 midterm exams (40%), 1 final exam (40%)

Final exam: Thursday, May 24 from 9:00 to 12:00 in the Ice Rink. Check registrar posting for updates.

Gradebook: managed on Stellar course web 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: Exams are closed book without cheat sheets, but this story sheet (which is basically what a cheat sheet for this course would look like if there were one) may help you study. 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 7): 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 9): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid)

  Problem Set One, due February 16 (students who have registered late may submit the first problem set on February 23)

• Lecture 3 (February 12): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (February 14): 2.3-2.4 Axioms of probability (see Paulos' NYT article and a famous hat problem)
• Lecture 5 (February 16): 2.5-2.7 Probability and equal likelihood (and a bit more history)

  Problem Set Two, due February 23

• Lecture 6 (February 20, Tuesday): 3.1-3.2 Conditional probabilities
• Lecture 7 (February 21): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (February 23): 4.1-4.2 Discrete random variables

  Problem Set Three, due March 2

• Lecture 9 (February 26): 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 (February 28): 4.5 Variance
• Lecture 11 (March 2): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.

  Practice Midterm Exam with partial solutions. 2009 Midterm One With Solutions. Spring 2011 midterm exam on Chapters 1-4 (plus 9.1) with solutions. Fall 2011 midterm exam 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 .

• Lecture 12 (March 5): 4.7 Poisson random variables
• Lecture 13 (March 7): REVIEW
• Lecture 14 (March 9): First midterm with solutions. First midterm with solutions.

  Problem Set Four, due March 16

• Lecture 15 (March 12): 9.1 Poisson processes
• Lecture 16 (March 14): 4.8-4.9 More discrete random variables
• Lecture 17 (March 16): 5.1-5.3 Continuous random variables

  Problem Set Five, due March 23

• Lecture 18 (March 19): 5.4 Normal random variables
• Lecture 19 (March 21): 5.5 Exponential random variables
• Lecture 20 (March 23): 5.6-5.7 More continuous random variables

  Problem Set Six, due April 6

• Lecture 21 (April 2): 6.1-6.2 Joint distribution functions
• Lecture 22 (April 4): 6.3-6.5 Sums of independent random variables
• Lecture 23 (April 6): 7.1-7.2 Expectation of sums

  Problem Set Seven, due April 13

• Lecture 24 (April 9): 7.3-7.4 Correlation and causation
• Lecture 25 (April 11): 7.5-7.6 Conditional expectation
• Lecture 26 (April 13): 7.7-7.8 Moment generating distributions

  Practice Midterm Exam Two with partial solutions and 2009 Midterm Two with solutions . Spring 2011 Second midterm exam 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.

• Lecture 27 (April 18): REVIEW
• Lecture 28 (April 20): Second midterm with solutions.

  Problem Set Eight, due April 27

• Lecture 29 (April 23): 8.1-8.2 Weak law of large numbers
• Lecture 30 (April 25): 8.3 Central limit theorem
• Lecture 31 (April 27): 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.

  Problem Set Nine, due May 4

• Lecture 32 (April 30): 9.2 Markov chains
• Lecture 33 (May 2): 9.3-9.4 Entropy
• Lecture 34 (May 4): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)

  Problem Set Ten, due May 11 (see this short martingale note for supplemental reading)

• Lecture 35 (May 7): More martingales
• Lecture 36 (May 9): Risk Neutral Probability and Black-Scholes (look up options quotes at the Chicago Board Options Exchange)
• Lecture 37 (May 11): REVIEW

• Lecture 38 (May 14:) REVIEW
• Lecture 39 (May 16): REVIEW

  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)

  Optional: song about identities (not all covered in this course) with practice problems at the end. If you want to work through this, watch a couple of times and try to get most of the formulas in short term memory. Then attempt the problems (each of which requires standard course concepts --- like additivity of expectation, indicators, order statistics, independence, multinomial coefficients, etc. --- along with one of the identities from the clip). The identities that appear in this clip and not elsewhere in the course (Riemann zeta function, Euler series, Ramanujan inverse pi formula, etc.) are not required knowledge for the final exam.

  Practice Final Problems (covering only later portion of the course) with partial solutions and Spring 2011 Final with solutions and Fall 2012 Final with solutions . Spring 2014 Final with solutions. Spring 2016 Final exam with solutions. Spring 2017 Final exam with solutions.

  Spring 2018 Final exam with solutions Thursday, May 24 from 9:00 to 12:00 in the Ice Rink