18.600 Probability and Random Variables: Spring 2016

Lectures: MWF 10-11 in 54-100

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

TAs: Cesar Cuenca and Hong Wang

Hong Wang's office hours:: Monday 4:30 to 6:30 in 2-231

Cesar Cuenca's office hours: Thursday 5:00 to 7:00 in 2-231

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" at Google, Amazon, or ebay. (Here's another free and fun book.)

Assignments: 10 problem sets (20%), 2 midterm exams (40%), 1 final exam (40%)

Final exam: Johnson Track Tuesday, May 17, 9-12. 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 last year 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.

TENTATIVE SCHEDULE

• Lecture 1 (February 3): 1.1-1.3 Permutations and combinations (also Pascal's triangle --- as studied (not invented) by Pascal, see also correspondence with Fermat).
• Lecture 2 (February 5): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid)

  Problem Set One, due February 12 (students who have registered late may submit the first problem set on February 19)

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

  Problem Set Two, due February 19

• Lecture 6 (February 16, Tuesday): 3.1-3.2 Conditional probabilities
• Lecture 7 (February 17): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (February 19): 4.1-4.2 Discrete random variables

  Problem Set Three, due February 26

• Lecture 9 (February 22): 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 24): 4.5 Variance
• Lecture 11 (February 26): 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

• Lecture 12 (February 29): 4.7 Poisson random variables
• Lecture 13 (March 2): REVIEW
• Lecture 14 (March 4): First midterm with solutions

  Problem Set Four, due March 11

• Lecture 15 (March 7): 9.1 Poisson processes
• Lecture 16 (March 9): 4.8-4.9 More discrete random variables
• Lecture 17 (March 11): 5.1-5.2 Continuous random variables

  Problem Set Five, due March 18

• Lecture 18 (March 14): 5.3 Uniform random variables
• Lecture 19 (March 16): 5.4 Normal random variables
• Lecture 20 (March 18): 5.5 Exponential random variables

  Problem Set Six, due April 1

• Lecture 21 (March 28): 5.6-5.7 More continuous random variables
• Lecture 22 (March 30): 6.1-6.2 Joint distribution functions
• Lecture 23 (April 1): 6.3-6.5 Sums of independent random variables

  Problem Set Seven, due April 8

• Lecture 24 (April 4): 7.1-7.2 Expectation of sums
• Lecture 25 (April 6): 7.3-7.4 Covariance and correlation. (Fun weekend activity: see how humans think about correlation and causation by typing "study" and "linked to" into google and paging through until you find 100 distinct "Study links A to B" headlines. How many do you think are real correlations (as opposed to statistical flukes or chance anomalies or measurement/methodology errors that might not appear in a larger, more careful study)? In how many cases do the authors provide what you need to assess (1) the strength of evidence for correlation existence (2) magnitude of the reported correlation (3) what is known about the plausibility of obvious causal and non-causal explanations?)
• Lecture 26 (April 8): 7.5-7.6 Conditional expectation

  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. There will be no formula sheet on the exam, but here is a story sheet which you can use to prepare in advance (but not during the exam itself; instead of memorizing, try to understand the stories well enough that remembering the formulas is automatic).

• Lecture 27 (April 11): 7.7-7.8 Moment generating distributions
• Lecture 28 (April 13): REVIEW
• Lecture 29 (April 15): Second midterm with solutions.

  Problem Set Eight, due April 22

• Lecture 30 (April 20): 8.1-8.2 Weak law of large numbers
• Lecture 31 (April 22): 8.3 Central limit theorem

  Problem Set Nine, due April 29

• Lecture 32 (April 25): 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 33 (April 27): 9.2 Markov chains
• Lecture 34 (April 29): 9.3-9.4 Entropy

  Problem Set Ten, due May 6 (see this short martingale note for supplemental reading)

• Lecture 35 (May 2): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)
• Lecture 36 (May 4): Risk Neutral Probability and Black-Scholes (look up options quotes at the Chicago Board Options Exchange)
• Lecture 37 (May 6): REVIEW

• Lecture 38 (May 9:) REVIEW
• Lecture 39 (May 11): REVIEW

  Problem outtakes (which for various reasons did not make it into a problem set, but which you can browse if you are curious)

  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 .

• Final exam (with solutions): Johnson Track, Tuesday, May 17, 9-12. Check registrar posting for updates.