Probability, Programming in R

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

1. Introduction to stochastic processes 1.1. Definition and basic concepts. 1.2. Types of processes. 2. Discrete-time Markov chains. 2.1. Definition and basic computations. 2.2. Classification of states. 2.3. Limiting and stationary distributions. 2.4. Limit theorems. 2.5. ML estimation of transition probabilities. 3. Markov chain Monte Carlo. 3.1. The Metropolis-Hastings algorithm. 3.2. The Gibbs sampler. 3.3. MCMC diagnosis. 4. Poisson processes. 4.1. Introduction. 4.2. The Poisson process. 4.2.1. Inter-arrival times. 4.2.2. Infinitesimal probabilities. 4.2.3. The connection with the uniform distribution. 4.2.4. Thinning and superposition 4.3. Non-homogeneous Poisson processes. 5. Continuous-time Markov chains 5.1. Introduction 5.2. Transition function and transition rates 5.3. Long-term behaviour 5.4. Time-reversibility 6. Brownian motion and Gaussian processes 6.1. Brownian Motion 6.2. Transformations and Properties 6.3. Extensions of the Brownian Motion 6.4. Gaussian processes

In each class, theoretical concepts are introduced. Numerical and simulated exercises are shown for better understanding. Applications are also made with real data.