Lecture 1 (click here)
- Course administration. Motivation for the course by Tomas Mc Kelvey (click here).
- Chapters 1 and 2.
Lecture 2 (click here)
- Chapters 3 and 4 (except 4.10, 4.12).
Lecture 3 (click here)
- Chapters 5.6-5.10 and 6.1-6.3.
Lecture 4 (click here)
- Chapters 6.4-6.5 and 6.8-6.9.
Lecture 5 (click here)
- Chapter 7.1-7.3.
- Tomas Mc Kelvey: MCMC.
Lecture 6 (click here)
- Chapters 7.4-7.9 and 8.
Lecture 7 (click here)
- Chapter 9.
Chapters 1 and 2
- Sample space, events as sets.
- Sigma-field, De Morgan laws.
- Probability measure, probability space.
- Conditional probability, independence.
- Pairwise independence is weaker than independence.
- Completeness and product spaces.
- Example: repeated coin tossing.
- Random variables as measurable functions.
- Bernoulli variables, the law of averages, Bernstein's inequality.
- Discrete and continuous distributions.
- Random vectors.
- Monte Carlo simulation.
Chapters 3 and 4 (except 4.10, 4.12)
- Discrete random variables, probability mass functions.
- Independence, Poisson flips.
- Expectation and its properties. Moments. Variance.
- Inclusion-exclusion formula.
- Binomial, Poisson, geometric, hypergeometric, negative binomial, multinomial distributions.
- Joint distributions, marginals. Independence of random variables.
- Covariance and correlation coefficient.
- Conditional distributions and conditional expectation.
- Sums of random variables. Simple random walk.
- The reflection principle. Hitting times. Arcsine laws.
- Continuous random variables, probability density functions.
- Uniform, exponential, normal, gamma, Cauchy, beta, Weibull distributions.
- Bivariate normal distribution.
- Cauchy-Schwartz inequality.
- Functions of random variables.
- Multivariate normal distribution.
- Sampling from a distribuion.
Chapters 5.6-5.10 and 6.1-6.3
- Expectation operator.
- Moment generating function and characteristic function. Examples.
- Weak law of large numbers.
- Central limit theorem.
- Markov chain, transition probabilities, Chapman-Kolmogorov equations.
- Recurrent (persistent) and transient, null-recurrent and positive states.
- Classification of chains.
Chapters 6.4-6.5 and 6.8-6.9
- Stationarity of Markov chains.
- Recurrence and null-recurrence.
- Mean recurrence times and the limits for the transition probabilities.
- Reversible chains, detailed balance equations.
- Birth processes and the Poisson process.
- Continuous time Markov chains.
- The generator matrix.
- Explosion of continuous time Markov chains.
- Jump chain.
Chapter 7.1-7.3
- Convergence a.s.
- Convergence in mean
- Convergence in probability
- Convergence in distribution
- Counterexamples
- Borel-Cantelli lemmas
- Dominated convergence theorem
Chapters 7.4-7.9 and 8
- Strong law of large numbers
- Law of the iterated logarithm
- Martingales
- Martingale convergence theorem
- Prediction and conditional expectation
Chapter 9
- Weakly and strongly stationary processes
- Autocovariance and autocorrelation functions
- Best linear predictor
- Spectral theorem for autocorrelation functions
- Discrete time stationary processes
- Spectral representation of stationary processes
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