Probability (Year 2 - Semester 2) Statistics (Year 3 - Semester 1) Stochastic Processes (Year 4 - Semester 1) - at least partial knowledge

K04: now the principles of probability calculus and statistical inference and how to apply them in solving real-life problems K06: Know the fundamental results of the theory of ordinary, partial differential and stochastic differential equations and their applications in mathematical models S03: Apply mathematical language and abstract-rigorous reasoning in the enunciation and demonstration of results in various areas of mathematics. S10: Apply the fundamentals of Bayesian statistics and computationally intensive techniques to implement Bayesian inference and prediction in machine learning. S13: Formulate real-world problems by means of mathematical models for their subsequent analysis and resolution. S14: Apply appropriate analytical or numerical techniques to solve mathematical models associated with real-world problems and interpret the results obtained. C06: Model real-world processes using stochastic processes and queuing theory, and simulate them on a computer. C07: Establish the definition of a new mathematical object, in terms of others already known for solving problems in different contexts.

* R programming and introduction to R Markdown * Probability refresher * Statistics refresher 1. Random numbers (Monte Carlo tecniques) 1.1 Probability and inference refresher 1.2 Statistical validation techniques 1.3 (Pseudo)random number generation 1.4 Approximation of probabilities and volumes 1.5 Monte Carlo integration 2. Simulating random variables and vectors 2.1 Inverse transform 2.2 Aceptance-rejection 2.3 Composition approach 2.4 Multivariate distributions 2.5 Multivariate normal distribution 3. Discrete event simulation 3.1 Poisson processes 3.2 Gaussian processes 3.3 Single- and multi-server Queueing systems 3.4 Inventory model 3.5 Insurance risk model 3.6 Repair problem 3.7 Exercising a stock option 4. Efficiency improvement (variance reduction) techniques 4.1 Antithetic variables 4.2 Control variates 4.3 Stratified sampling 4.4 Importance sampling 5. MCMC 5.1 Markov chains 5.2 Metropolis-Hastings 5.3 Gibbs sampling 6. Introduction to the bootstrap 6.1 The bootstrap principle 6.2 Estimating standard errors 6.3 Bootstrap Inference (Confidence Intervals) 6.4 The two-sample problem 6.5 Bootstrapping regression models

- Lectures and problem sessions with a computer: introducing the theoretical concepts and developments with examples, and solving problems: 25 on-site hours - Homework: 49 non on-site hours - Evaluation sessions (continuous evaluation and final exam): 5 on-site hours - Specific exam preparation: 49 non on-site hours