Probability (Year 2 - Semester 2)

- Understand the theoretical foundations and basic properties of Stochastic Processes. - Learn the basic simulation techniques for the studied models. - Solve problems based on the studied models.

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. 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.

1. Introduction to Stochastic Processes. 1.1. Basic Definitions and Notations. 1.2. Examples: branching processes and queues. 1.3. Review of Conditional Expectation. 1.4. Review of Characteristic Functions and applications. 2. Discrete time Markov Chains. 2.1. Basic Definitions and Notations. 2.2 Chapman-Kolmogorov Equations and classification of states. 2.3. Asymptotic results. 2.4. First Step Analysis. 2.5. Random Walks and Success Runs. 2.6 The Geo/Geo/1 queue. 3. Renewal Theory and Poisson process. 3.1 Definition and basic notions. 3.2 The Elementary Renewal Theorem.¿ 3.3 The Key Renewal Theorem. 3.4 The Delayed Renewal Theorem. 3.5 Compound Poisson Process. 4. Continuous time Markov Chains. 4.1 Definition and basic notions¿ 4.2 Chapman-Kolmogorov Equations and Limit Theorems 4.3 Birth and Death Processes (M/M/m queues). 5. Continuous time Markov Processes: Brownian Motion. 5.1 Brownian Motion and Gaussian Processes. 5.2 Variations and Extensions. 5.3 Hitting times.¿ 5.4 Relation with Martingales.

- Clases magistrales: Presentación de conceptos, desarrollo de la teoría y ejemplos, 2.2 ECTS - Clases de resolución de problemas: 2.2 ECTS - Prácticas de ordenador: 0.6 ECTS - Sesiones de evaluación (exámenes de evaluación continua y examen final): 1 ECTS