Theory of Probability (Fall 2023)

Lectures: Tuesdays, Thursdays 11:00-12:15PM 194M 304
Office hours: Thursdays 3:00-5:00PM WWH 1013
Recitation: Fridays 11:00-12:15 room 102 19W 4th
TA: Ahmet Keles (aak10037)

Theory of Probability (Math 233) is an introduction to probability theory.

Schedule

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.

Textbook

The main textbook for the course is Ross, A First Course in Probability, 10th Ed., 2018. An earlier edition of the textbook is available for free via NYU libraries. It is not necessary to have the latest edition.

Grading

Homework 20%
Midterm exam with lower score 15%
Midterm exam with higher score 25%
Final exam 40%

Midterm 1: October 3
Midterm 2: November 7

Lectures

  • Counting, permutations, combinations
    Ross 1.1-1.4
  • Multinomial coefficients, sample spaces
    Ross 1.5, 2.1-2.2
  • Axioms of probability, some propositions
    Ross 2.3-2.4
  • Sample spaces with equally likely outcomes, probability as belief
    Ross 2.5, 2.7
  • Conditional probability, Bayes' rule
    Ross 3.1-3.3
  • Independent events
    Ross 3.4
  • Conditional probably, again
    Ross 3.5
  • Discrete random variables, expected value
    Ross 4.1-4.3
  • Exam 1
  • Expectations of functions of a RV; variance
    Ross 4.4-4.5
  • Bernoulli and binomial random variables
    Ross 4.6
  • Poisson random variables
    Ross 4.7
  • Other discrete random variables, expectations of sums; Properties of the CDF
    Ross 4.8-4.10
  • Continuous random variables
    Ross 5.1-5.2
  • Uniform and normal random variables
    Ross 5.3-5.4
  • Normal approximation to the binomial distribution Exponential, Gamma, and Cauchy distributions
    Ross 5.4.1, 5.5-5.6
  • Functions of continuous random variables; joint distributions
    Ross 5.7, 6.1
  • Distribution of sums of independent random variables
    Ross 6.2-6.3
  • Exam 2
  • Conditional distributions
    Ross 6.4-6.5
  • Order statistics; functions of several random variables
    Ross 6.6-6.7
  • Expectations of sums, covariance, correlation
    Ross 7.1-7.2, 7.4
  • Conditional expectation and prediction
    Ross 7.5-7.6
  • Moment generating functions; properties of normal variates
    Ross 7.7-7.8
  • Central Limit Theorem
    Ross 8.1-8.3
  • Strong Law of Large Numbers, more inequalities Ross 8.4-8.5
  • Poisson processes
    Ross 9.1
  • Final Exam

References

  • Ross, A First Course in Probability, 10th Ed., 2018