Probability - The Science of Uncertainty and Data

Unit 1: Probability models and axioms

• Probability models and axioms
• Mathematical background: Sets; sequences, limits, and series; (un)countable sets.

Unit 2: Conditioning and independence

• Conditioning and Bayes' rule
• Independence

Unit 3: Counting

• Counting

Unit 4: Discrete random variables

• Probability mass functions and expectations
• Variance; Conditioning on an event; Multiple random variables
• Conditioning on a random variable; Independence of random variables

Unit 5: Continuous random variables

• Probability density functions
• Conditioning on an event; Multiple random variables
• Conditioning on a random variable; Independence; Bayes' rule

Unit 6: Further topics on random variables

• Derived distributions
• Sums of independent random variables; Covariance and correlation
• Conditional expectation and variance revisited; Sum of a random number of independent random variables

Unit 7: Bayesian inference

• Introduction to Bayesian inference
• Linear models with normal noise
• Least mean squares (LMS) estimation
• Linear least mean squares (LLMS) estimation

Unit 8: Limit theorems and classical statistics

• Inequalities, convergence, and the Weak Law of Large Numbers
• The Central Limit Theorem (CLT)
• An introduction to classical statistics

Unit 9: Bernoulli and Poisson processes

• The Bernoulli process
• The Poisson process
• More on the Poisson process

Unit 10 (Optional): Markov chains

• Finite-state Markov chains
• Steady-state behavior of Markov chains
• Absorption probabilities and expected time to absorption