| 1 |
Introduction. Basics of Probability (sample space, events, sigma-algebra of measurable sets, measurable space, measure, probability measure, non-measurable sets) |
| 2 |
Random variables (discrete and continuous), indicators, distribution, expectation, absolute convergence, mean, variance, covariance, conditional probability, independence, law of total probability |
| 3 |
Bernoulli process, binomial, geometric, negative binomial, convex functions, strict convexity, Jensen’s inequality, entropy, cross-entropy, KL divergence |
| 4 |
KL divergence (continued), hypergeometric distribution, exchangeability, tail sum formula |
| 5 |
More on exchangeability and hypegeometric distribution, Boole’s inequality (union bound), Cauchy-Schwarz inequality, Markov’s inequality, Chebyshev’s inequality, weak LLN |
| 6 |
Convergence of Random Variables (almost sure convergence, convergence in probability, convergence in r-mean, convergence in distribution), strong LLN, CLT, equal in distribution vs equal in probability, examples of convergence |
| 7 |
Convergence of Random Variables (continued), relations between different convergence concepts, moment generating function, applications to show convergence in distribution (from Geometric to Exponential distribution; from Negative Binomial to Gamma distribution) |
| 8 |
Sums of random variables, convolution, empirical cumulative distribution function, Glivenko-Cantelli Theorem, Chernoff Inequalities, bounded random variables, Hoeffding’s inequality |
| 9 |
Proof of Hoeffding’s inequality, sub-Gaussian random variables, Hoeffding’s lemma, entropy approximation of Binomial, normal approximation of Binomial, Proof of CLT for sums of Bernoulli random variables |
| 10 |
A sum of independent normal RVs, application of the CLT, proof of the CLT, Berry-Eseen Theorem, Extensions of CLT |
| 11 |
Characteristic function, Levy’s continuity theorem, from binomial to Poisson, multinomial distribution, Poissonization of the multinomial, probability generating function (PGF), compounding |
| 12 |
Transformation of a single RV, invertible vs. non-invertible transformations, one-to-one differentiable transformations, many-to-one transformations, Cauchy distribution, log-normal distribution, chi-square/gamma distribution, chi distribution, quantile transform, quantile function |
| 13 |
Bivariate jointly continuous random variables, bivariate normal, marginal densities, polar coordinates, linear transformations, invertible linear transformations, rotations, orthogonal transformations |
| 14 |
Joint density, independence, review of invertible affine transformations, general invertible transformations, gamma and beta distributions |
| 15 |
Beta distribution, Dirichlet distribution, order statistics, CDF and PDF of the \(j\)th order statistics, uniform order statistics, general order statistics via quantile transformation, joint distribution of order statistics, gaps (spacings), exchangeability |
| 16 |
Linear models, MLE of regression coefficients, hypothesis testing, conditional densities of continuous RVs, independence of continuous RVs, Law of Total Probability for continuous RVs, Cauchy, t-distribution, Bayes Rule for continuous RVs |
| 17 |
Bayesian inference, conjugate priors for binomial and multinomial distributions, conjugate priors for Normal random variables, model selection, prior odds, Bayes factor, posterior odds |
| 18 |
Conditional expectation, law of total expectation (aka law of iterated expectation, tower property), Wald’s identity, statistical risk minimization, MSE, MAE |
| 19 |
Jointly normal/Gaussian RVs (a.k.a., multivariate normal), covariance matrices, PDF of multivariate normal distribution, bivariate normal example, marginal distribution of multivariate normal, conditional distribution of multivariate normal |
| 20 |
Positive (semi-)definite matrices, moment generating function for multivariate normal, idempotent matrices, relation between multivariate normal and chi-square distributions, multivariate CLT, Pearson’s chi-square test |
| 21 |
Guassian processes, covariance function (kernel), stationary kernels, isotropic kernels, sampling from GP, regularization (nugget), radial basis function, Matérn kernel, Browning motion, Brownian bridge, Ornstein-Uhlenbeck kernel, linear kernel, predictions from GP |
| 22 |
Proof of the Glivenko-Cantelli theorem (weak version), convergence of the empirical process to the Brownian bridge, Kolmogorov-Smirnov distribution |
| 23 |
Stochastic/random process, Galton-Watson branching process \(\{X_t, t\in \mathbb{N}_0\}\), offspring distribution, \(\mathbb{E}[X_t]\), \(\text{Var}[X_t]\), super-criticality, criticality, sub-criticality, extinction probability via probability generating function, extinction time distribution |
| 24 |
Discrete-time Markov chain, transition probability, homogeneity, \(n\)-step transition probability, Chapman-Kolmogorov equation, classification of states, irreducible Markov chain, first passage probability, return probability, mean recurrence time, recurrent/persistent states, transient states, first-step analysis, hitting probability, hitting time, stationary distribution, existence and uniqueness |
| 25 |
The long-term behavior of Markov chains, period, regular/ergodic Markov chains, limiting distributions, WLLN for irreducible finite Markov chains, ergodic theorem, fundamental matrix of irreducible Markov chains, mean first passage times, reversed process, reversibility, detailed balance |
| 26 |
Continuous-time Markov chain, standard transition matrices, generator, stable Markov chains, conservative Markov chains, Kolmogorov Forward Equation, Kolmogorov Backward Equation, holding time distributions, Poisson Process, Birth-Death processes, jump process/jump chain/embedded chain |