18.440 Probability and Random Variables: Spring, 2011

Lectures: MWF 11-12, 4-370

Office hours: Wednesday 1-3, 2-180

TA: Dimiter Ostrev

TA Office hours: Thursdays 12:30 to 2:30, 2-229

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

Assignments: Homeworks (20%), midterm exams (40%), final exam (40%)

Stellar course web site

TENTATIVE SCHEDULE

• Lecture 1 (February 2): 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 4): 1.4-1.5 Multinomial coefficients and more counting

  Problem Set One, due February 11

• Lecture 3 (February 7): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (February 9): 2.3-2.4 Axioms of probability (see Paulos' NYT article and a famous hat problem) )
• Lecture 5 (February 11): 2.5-2.7 Probability and equal likelihood (and a bit more history )

  Problem Set Two, due February 18

• Lecture 6 (February 14): 3.1-3.2 Conditional probabilities
• Lecture 7 (February 16): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (February 18): 4.1-4.2 Discrete random variables

  Problem Set Three, due February 25

• 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 23): 4.5 Variance
• Lecture 11 (February 25): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.

  Problem Set Four, due March 4

• Lecture 12 (February 28): 4.7 Poisson random variables
• Lecture 13 (March 2): 9.1 Poisson processes
• Lecture 14 (March 4): 4.8-4.9 More discrete random variables

  Practice Midterm Exam with partial solutions (here is an old midterm and see also Jonathan Kelner's old 18.440 pages and 2009 Midterm One With Solutions )

• Lecture 15 (March 6): 5.1-5.2 Continuous random variables
• Lecture 16 (March 9): REVIEW
• Lecture 17 (March 11): MIDTERM EXAM on CHAPTERS 1-4 (plus 5.1, 5.2, and 9.1) with solutions

  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
• Lecture 26 (April 8): 7.5-7.6 Conditional expectation

  Practice Midterm Exam Two with partial solutions and 2009 Midterm Two with solutions .

• Lecture 27 (April 11): 7.7-7.8 Moment generating distributions
• Lecture 28 (April 13): REVIEW
• Lecture 29 (April 15): Second midterm exam on 1-7 (plus 9.1) 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 (plus martingale note)

• 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

  Practice Final with partial solutions

• (Week of May 16-20): Final Exam