Date

Content

9/16

Lecture 1. Highlights of the syllabus. Multiplication rule, permutation, combination, and addition rule.

9/18

Lecture 2. Particles in a box, axioms of probability, set operations, experiments with equally  likely outcomes

9/19

Lecture 3 & Quiz 0. Story proofs. How to write proofs.

9/23

Lecture 4. Inclusion-exclusion, conditional probability, and derangement.

9/30

Lecture 5 & Quiz 1. Independence, Bayes' rule, and the birthday problem (not including joint independence).

10/2

Lecture 6. Law of total probability, Multiple choice problem, Monty Hall, Gambler’s ruin.

10/7

Lecture 7 & Quiz 2. Independence revisited, all named problems revisited, and random variables

10/9

Lecture 8. Random variables & vectors, support, CDF, PMF, Discrete uniform, Binomial, Bernoulli, Geometric

10/14

Lecture 9 & Quiz 3. Joint PMF, independence, and conditional PMF. Expectation, transformations, the law of the unconscious statistician.

10/16

Lecture 10. linearity of expectations, variance, sample mean, law of large numbers, central limit theorem (preview), and the Gaussian distribution. 

10/21

Lecture 11 & Quiz 5. Coupon/Labubu collector problem. The intuitions for the Law of Large Numbers and Central Limit Theorem (revisit). 68-97-99.7 rule, and the Normal distribution in general. 

10/23

Lecture 12. Probability density function. Uniform distribution. Exponential distribution. Joint CDF and joint PDF.

10/28

Lecture 13 & Quiz 6. Expectation and variance of continuous random variables. Memory-less property of Exponential. High-level ideas of Poisson distribution, Poisson process, and Gamma distribution. Joint distributions (joint, marginal, conditional) 

10/30

Lecture 14. Joint distributions. High-level ideas of order statistics, and exercises: symmetry of iid random variables, probability of forming a triangle, etc. 

11/4

Lecture 15 & Quiz 7. Moment generating functions. Central Limit Theorem (re-revisit and proof).

11/6

Lecture 16. Conditional expectation. Law of total expectation. 

11/11

Lecture 17 & Quiz 8. Covariance and correlation. END of exam materials. Maximum and minimum of iid continuous random variables.

11/13

Lecture 18. Example: joint distribution of the maximum and minimum of two iid continuous random variables. ‘Choosing the best candidate’ problem. 

11/18

Lecture 19 & Quiz 9. Review: estimating pi, the birthday problem, and Gambler’s Ruin.

11/20

Lecture 20 (optional). Final review using the chicken-egg problem and the ‘everything everywhere all at once’ problem.