· 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