Supervised 3 groups in Easter 2025.

Syllabus

Part 1 - Introduction to Probability Introduction. Counting/Combinatorics (revision), Probability Space, Axioms, Union Bound. Conditional probability. Conditional Probabilities and Independence, Bayes’Theorem, Partition Theorem

Part 2 - Discrete Random Variables Random variables. Definition of a Random Variable, Probability Mass Function, Cumulative Distribution, Expectation. Probability distributions. Definition and Properties of Expectation, Variance, different ways of computing them, Examples of important Distributions (Bernoulli, Binomial, Geometric, Poisson), Primer on Continuous Distributions including Normal and Exponential Distributions. Multivariate distributions. Multiple Random Variables, Joint and Marginal Distributions, Independence of Random Variables, Covariance.

Part 3 - Moments and Limit Theorems Introduction. Law of Average, Useful inequalities (Markov and Chebyshef), Weak Law of Large Numbers (including Proof using Chebyshef’s inequality), Examples. Moments and Central Limit Theorem. Introduction to Moments of Random Variables, Central Limit Theorem (Proof using Moment Generating functions), Example.

Part 4 - Applications/Statistics Statistics. Classical Parameter Estimation (Maximum-Likelihood-Estimation, bias, sample mean, sample variance), Examples (Collision-Sampling, Estimating Population Size). Algorithms. Online Algorithms (Secretary Problem, Odd’s Algorithm).