Syllabus
· Events and probabilities i. Axiomatic definition of probability ii. Conditional probability, independence and Bayes’ Rule iii. Random variables and univariate distribution functions iv. Functions and transformation of random variables v. Moments and moment generating functions
· Univariate probability distributions i. Discrete distributions: Bernoulli, binomial, negative binomial, (hyper)geometric, Poisson ii. Continuous distributions I: normal, lognormal, logistic, Cauchy, Laplace iii. Continuous distributions II: Beta, Gamma and their special cases, F, Student’s t, Pareto iv. Continuous distributions III: extreme values: Gumbel, Fréchet and (reverse) Weibull
· Multivariate probability distributions i. Random vectors, joint distributions, marginal distributions, transformations ii. Independence of random variables and vectors, random products and random ratios iii. Moments of random vectors, covariance, correlation iv. Multivariate moment generation, sum of independent random variables v. Conditional distributions and moments, Law of Iterated Expectations, Law of Total Variance vi. Key multivariate distributions: multinomial, multivariate normal
· Samples and sample statistics i. Samples, random samples and their properties ii. Sampling from the univariate and multivariate normal distributions iii. Order statistics and some key associated results iv. The sufficiency principle and sufficient statistics
· Estimation and Inference i. Point estimation: the method of moments and maximum likelihood estimation ii. Evaluating estimators: loss functions, unbiasedness, consistency, the Cramér-Rao bound iii. Inference: tests of hypotheses, and analysis of selected exact results iv. Inference: interval estimation and analysis of selected exact results
· Introduction to asymptotic theory i. Random sequences, convergence in probability, almost sure convergence ii. Properties of convergent sequences, Laws of Large Numbers, and implications for estimation iii. Convergence in distribution, Slutsky’s Theorem, and the Cramér-Wold device iv. Central Limit Theorems, the Delta Method, and implications for estimation
· Linear Projections and Regression i. Linear socio-economic relationships: some classical examples ii. Linear predictors, linear projections and conditional expectation iii. The least squares estimator: derivation and algebraic properties iv. Introduction to the linear regression model, dummy variables