An elementary course of Probability and Statistics

To acquire basic rudiments of the theory of stochastic processes. Modeling real problems through Markov processes and Martingales. To solve problems using the appropriate stochastic methodologies and techniques.

1. Discrete-time Markov chains - Definition and basic computations - Classification of states - Limiting and stationary distributions - Limit theorems - ML estimation of transition probabilities 2. Markov chain Monte Carlo - The Metropolis-Hastings algorithm - The Gibbs sampler - MCMC diagnosis 3. Poisson process - Definition - Inter-arrival times - Infinitesimal probabilities - The connection with the uniform distribution - Thinning and superposition - Non-homogeneous Poisson processes 4. Continuous-time Markov chains - Introduction - Transition function and transition rates - Long-term behaviour - Time-reversibility 5. Brownian motion and Gaussian processes - Brownian Motion - Transformations and Properties - Extensions of the Brownian Motion - Gaussian processes

Every week there is a class. In each class, the theoretical concepts are usually introduced, numerical and simulated exercises are shown to better understand them and examples of models that can be used in more specific applications are made.