18.440 Probability and Random Variables: Fall, 2009

Lectures: MWF 11-12, 4.370

Office hours: Monday 1-2 and Wednesday 2-3, 2-180

TA: Zhenqi He

TA Office hours: Thursday 4-6, 2-102

Text: A First Course in Probability, 8th edition, by Sheldon Ross

Assignments: Homeworks (10%), midterm exams (40%), final exam (50%)

Stellar course web site

TENTATIVE SCHEDULE

• Lecture 1 (September 9): 1.1-1.3 Permutations and combinations (also Pascal's triangle --- as studied (not invented) by Pascal, see also correspondence with Fermat. )
• Lecture 2 (September 11): 1.4-1.5 Multinomial coefficients and more counting

  Problem Set One, due September 18

• Lecture 3 (September 14): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (September 16): 2.3-2.4 Axioms of probability
• Lecture 5 (September 18): 2.5-2.7 Probability and equal likelihood (and a bit more history and a famous hat problem)

  Problem Set Two, due September 25

• Lecture 6 (September 21): 3.1-3.2 Conditional probabilities
• Lecture 7 (September 23): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (September 25): 4.1-4.2 Discrete random variables

  Problem Set Three, due October 2

• Lecture 9 (September 28): 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 30): 4.5 Variance
• Lecture 11 (October 2): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.

  Problem Set Four, due October 9

• Lecture 12 (October 5): 4.7 Poisson random variables
• Lecture 13 (October 7): 9.1 Poisson processes
• Lecture 14 (October 9): 4.8-4.9 More discrete random variables

  Practice Midterm Exam (here is an old midterm and see also Jonathan Kelner's old 18.440 pages)

• Lecture 15 (October 13): REVIEW
• Lecture 16 (October 14): MIDTERM EXAM on CHAPTERS 1-4 (plus 9.1)

  Midterm One With Solutions

• Lecture 17 (October 16): 5.1-5.2 Continuous random variables
• Lecture 18 (October 19): 5.3 Uniform random variables
• Lecture 19 (October 21): 5.4 Normal random variables
• Lecture 20 (October 23): 5.5 Exponential random variables

  Problem Set Five, due October 30

• Lecture 21 (October 26): 5.6-5.7 More continuous random variables
• Lecture 22 (October 28): 6.1-6.2 Joint distribution functions
• Lecture 23 (October 30): 6.3-6.5 Sums of independent random variables

  Problem Set Six, due November 6

• Lecture 24 (November 2): 7.1-7.2 Expectation of sums
• Lecture 25 (November 4): 7.3-7.4 Covariance
• Lecture 26 (November 6): 7.5-7.6 Conditional expectation

  Practice Problem Set, finish by November 13

• Lecture 27 (November 9): 7.7-7.8 Moment generating distributions

  Practice Midterm Exam Two (and another source for old exam reviews)

• Lecture 28 (November 13): REVIEW
• Lecture 29 (November 16): MIDTERM EXAM on 1-7 (plus 9.1)

  Midterm Two with solutions

• Lecture 30 (November 18): 8.1-8.2 Weak law of large numbers
• Lecture 31 (November 20): 8.3 Central limit theorem

  Problem Set Seven, due November 30

• Lecture 32 (November 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 33 (November 25): 9.2 Markov chains

  Problem Set Eight, due December 4 (plus martingale note)

• Lecture 34 (November 30): 9.3-9.4 Entropy
• Lecture 35 (December 2): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)
• Lecture 36 (December 4): Risk Neutral Probability and Black-Scholes (look up options quotes at the Chicago Board Options Exchange)
• Lecture 37 (December 7): REVIEW
• Lecture 38 (December 9): REVIEW

  Practice Final

• (Week of December 14-18): Final Exam