18.440 Probability and Random Variables: Spring, 2014

Lectures: MWF 11-12, 54-100

Office hours: Wednesday 4 to 6, E17-312

TA: Dimiter Ostrev

TA office hours: Thursday 3-5, E17-301W

Text: A First Course in Probability, 8th edition, by Sheldon Ross. (Here's another free and fun-to-read book you might enjoy.) I am using the 8th edition, but students are welcome to use 6th, 7th, or 9th editions as well. Both hard copies and electronic versions can be found online (e.g., by searching for Sheldon Ross probability at amazon.com or google.com or ebay.com...)

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

Final exam on Tuesday, May 20 from 1:30 to 4:30 PM on Johnson Track

Stellar course web site

TENTATIVE SCHEDULE

• Lecture 1 (February 5): 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 7): 1.4-1.5 Multinomial coefficients and more counting (see Pascal's pyramid)

  Problem Set One, due February 14

• Lecture 3 (February 10): 2.1-2.2 Sample spaces and set theory
• Lecture 4 (February 12): 2.3-2.4 Axioms of probability (see Paulos' NYT article and a famous hat problem)
• Lecture 5 (February 14): 2.5-2.7 Probability and equal likelihood (and a bit more history )

  Problem Set Two, due February 24

• Lecture 6 (February 18, Tuesday): 3.1-3.2 Conditional probabilities
• Lecture 7 (February 19): 3.3-3.5 Bayes' formula and independent events
• Lecture 8 (February 21): 4.1-4.2 Discrete random variables

  Problem Set Three, due February 28

• Lecture 9 (February 24): 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 26): 4.5 Variance
• Lecture 11 (February 28): 4.6 Binomial random variables, repeated trials and the so-called Modern Portfolio Theory.

  Problem Set Four, due March 7

• Lecture 12 (March 3): 4.7 Poisson random variables
• Lecture 13 (March 5): 9.1 Poisson processes
• Lecture 14 (March 7): 4.8-4.9 More discrete random variables

  Practice Midterm Exam with partial solutions (here is an old midterm and 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 , and Fall 2012 midterm with solutions.

• Lecture 15 (March 10): 5.1-5.2 Continuous random variables
• Lecture 16 (March 12): REVIEW
• Lecture 17 (March 14): First Midterm with solutions

  Problem Set Five, due March 21

• Lecture 18 (March 17): 5.3 Uniform random variables
• Lecture 19 (March 19): 5.4 Normal random variables
• Lecture 20 (March 21): 5.5 Exponential random variables

  Problem Set Six, due April 4

• Lecture 21 (March 31): 5.6-5.7 More continuous random variables
• Lecture 22 (April 2): 6.1-6.2 Joint distribution functions
• Lecture 23 (April 4): 6.3-6.5 Sums of independent random variables

  Problem Set Seven, due April 11

• Lecture 24 (April 7): 7.1-7.2 Expectation of sums
• Lecture 25 (April 9): 7.3-7.4 Covariance and correlation. REMARK: Everyone knows that correlation does not imply causation... or do they? Try typing "study" and "linked to" into google and page through until you find 100 distinct headlines roughly of the form "Study links A to B". How many seem to be real correlations (as opposed to statistical flukes or chance anomalies or measurement/methodology errors that might not appear in a larger, more careful study)? Do authors the provide the info 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 the most obvious causal and non-causal explanations?
• Lecture 26 (April 11): 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 and Fall 2012 second midterm with solutions

• Lecture 27 (April 14): 7.7-7.8 Moment generating distributions
• Lecture 28 (April 16): REVIEW
• Lecture 29 (April 18): Second Midterm with solutions

  Problem Set Eight, due April 25

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

  Problem Set Nine, due May 2

• Lecture 32 (April 28): 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 30): 9.2 Markov chains
• Lecture 34 (May 2): 9.3-9.4 Entropy

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

• Lecture 35 (May 5): Martingales and the Optional Stopping Time Theorem (see also prediction market plots)
• Lecture 36 (May 7): Risk Neutral Probability and Black-Scholes (look up options quotes at the Chicago Board Options Exchange)
• Lecture 37 (May 9): REVIEW

• Lecture 38 (May 12:) REVIEW
• Lecture 39 (May 14): REVIEW

  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 .

• May 20: Final exam on Johnson Track from 1:30 to 4:30 PM... and here is the actual Spring 2014 Final with solutions .