Probability and Statistics

This course provides a rigorous introduction to discrete-time Markov chains (DTMCs), a fundamental class of stochastic processes with broad applications in science, engineering, economics, and data science. Topics include the Markov property, transition matrices, classification of states (recurrent, transient, absorbing), stationary and limiting distributions, absorption probabilities, and the concept of reversible distributions. The course explores key examples, such as a two-state DTMC and the gambler’s ruin problem in both symmetric and asymmetric settings. In the final part, we introduce criteria for establishing the existence, uniqueness, and convergence of stationary distributions, considering both finite and countable state spaces. References: 1) Ross, Pekoz: A Second Course in Probability (Chapter 5) 2) Norris: Markov Chains

MODULE 2 covers the following topics: 2.1) EXPLORATORY DATA ANALYSIS - summarising quantities (sample mean/median/mode/variance) - box-plot, scatter plot, histograms, stem-and-leaf 2.2) ESTIMATION - Goal of statistical analyses - method of moments - maximum likelihood estimation - confidence intervals 2.3) PROPERTIES OF THE ESTIMATORS - consistency - distorsion - UMVUE - sufficient statistics 2.4) INTRODUCTION TO BAYESIAN STATISTICS - Bayes Theorem - choice of the prior and its influence - conjugate priors - Bayesian estimators - Markov Chain Monte Carlo simulation 2.5) BAYESIAN STATISTICS - credible intervals - hypothesis testing - Bayes factor - hierarchical models Most of the module 2 has been based on the knowledge of Professor Fabrizio, without a book, but he also suggested the following two references: Probability and Statistics: The Science of Uncertainty Michael J. Evans and Jeffrey S. Rosenthal University of Toronto available online at https://www.utstat.toronto.edu/mikevans/jeffrosenthal/ Bayesian Data Analysis Third edition Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson,  Aki Vehtari, Donald B. Rubin available online at https://sites.stat.columbia.edu/gelman/book/BDA3.pdf