Elements of probability theory: probability spaces, random variables, expected value, variance, moments. Binomial variables, Poisson variables, normal variables and others. Conditional probability, theorem of Bayes, independent events and random variables, conditional expectation. Chebyshev's inequality and concentration inequalities, law of large numbers, central limit theorem.
Introduction to measure theory: Lebesgue measure, Lebesgue integral, abstract measures and probability measures. Introduction to stochastic processes, covariance and correlation, martingales, Brownian motion.
Elements of statistics: sampling, confidence intervals, test of hypotheses and test statistics, consistent estimators, unbiased estimators, estimators of the mean and variance, Student's T-distribution and the chi^2 distribution, test of goodness of fit, regression, maximum likelihood.