Probbability (Year 2 - Semester 2)

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