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
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