Probability

This introductory lecture establishes the naive definition of probability, emphasizing the foundational role of counting. It covers the multiplication rule, binomial coefficients, and the sampling table for various replacement and order scenarios.

This lecture introduces combinatorial 'story proofs' as a method for establishing identities like Vandermonde's identity, emphasizing intuitive insight over algebraic manipulation. It then lays the foundational axioms of probability theory, including discussion of Bose-Einstein sampling.

We discuss the birthday problem (how many people do you need to have a 50% chance of there being 2 with the same birthday?), the matching problem (de Montmort), inclusion-exclusion, and properties of probability.

We introduce conditional probability, independence of events, and Bayes' rule.

We continue further with conditional probability, and discuss the law of total probability, the so-called prosecutor's fallacy, a disease testing example, and the crucial distinction between independence and conditional independence.

We show how conditional probability sheds light on two of the most famous puzzles in statistics, both of which are often counterintuitive (at first): the Monty Hall problem and Simpson's paradox.

This lecture analyzes the classic Gambler's Ruin problem, modeling a finite-state absorbing Markov chain. It then introduces the fundamental concept of random variables and details the Bernoulli and Binomial probability distributions.

This lecture introduces the foundational concepts of random variables and their distributions. It formally defines cumulative distribution functions (CDFs) and probability mass functions (PMFs), culminating in an analysis of the Hypergeometric distribution.

This lecture introduces expected values, exploring indicator random variables and the principle of linearity to calculate means. It also establishes the fundamental bridge between probability and expectation and presents the Geometric distribution.

This session rigorously proves linearity of expectation and applies it to solve a Putnam problem. It introduces the Negative Binomial distribution and analyzes the intriguing St. Petersburg Paradox.

This lecture introduces the Poisson distribution, highlighting its significance as a fundamental discrete distribution in statistics. It examines its applications as an approximation and its relationship with the Binomial distribution.

This lecture delineates the distinctions between discrete and continuous probability distributions, introducing probability density functions (PDFs). It also covers variance, standard deviation, and the Uniform distribution.

This session introduces the Normal distribution, emphasizing its pervasive importance and widespread application across various fields of statistics. It establishes its foundational role in probabilistic modeling.

This lecture examines location and scale parameters in distributions, along with standardization techniques. It thoroughly explains the Law of the Unconscious Statistician (LOTUS) and applies it to derive the variance of a Poisson distribution.

This lecture introduces the Exponential distribution, which is distinctly characterized by its memoryless property. The session, though concise due to midterm discussions, covers the core aspects of this crucial continuous distribution.

This session introduces moment generating functions (MGFs), detailing their diverse applications in probability theory. It also discusses Laplace's rule of succession and the hybrid form of Bayes' rule.

Building on prior knowledge, this lecture employs MGFs to derive moments for Exponential and Normal distributions and determine the distribution of sums of Poissons. It also initiates the discussion on joint distributions.

This lecture elaborates on joint, conditional, and marginal distributions, introducing the 2-D LOTUS. It establishes the condition E(XY)=E(X)E(Y) for independent variables and solves the expected distance and chicken-egg problems.

This lecture introduces the Multinomial distribution, highlighting its properties and applications as a key multivariate discrete distribution. It also includes an example involving the Cauchy distribution.

This session introduces the concepts of covariance and correlation, demonstrating their roles in measuring variable relationships. It shows how to calculate the variance of a sum, including for a Hypergeometric random variable.

This lecture explores transformations of random variables (change of variables) and introduces the LogNormal distribution. It covers convolutions for sums of variables and demonstrates how probability can be used to prove existence.