Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 0 |
As a general requirement, in order to be able to follow this subject, good knowledge is required at the practical level of analysis and calculus or, more specifically, integration and series. As more specific requirements, it is necessary to have previously studied a course in Probability and Stochastic Modeling.
The aim of this subject is, on the one hand, to introduce the student in the part of the theory of probability called theory of stochastic processes, whose purpose is to study the random phenomena that evolve over time or in the space We will see the basic generalities of these models and we will study some specific models.
The discrete Markov chains will be studied in general and in the particular case of the random walk. We will also study the continuous-time Markov chains, such as the Poisson process or the birth and death processes. Finally we will also introduce the Brownian motion.
1. Discrete-time Markov chains.
1.1. Motivation: the random walk.
1.2. Definitions. Basic properties. Transition matrix.
1.3. Stoping time. Strong Markov property.
1.4. Recurrence and transience.
1.5. Asymptotic behavior. Invariant distribution
1.6. Ergodic theorem
1.7. More aspects about random walk
2. Continuous-time Markov chains.
2.1. Motivation: the Poisson process.
2.2. Basic properties. Generating matrix. Recurrence and transience.
2.3. Invariant distribution
2.4. Ergodic theorem
2.5. More aspects of the Poisson process.
3. The Brownian motion.
This subject is semiannual and consists of two hours of theory and one hour of problems per week. There will also be three two-hour seminars.
In theory classes, the teacher plays the main role and we will work with the structure definition-theorem-proof-application.
In the problem classes students will leave on the board to solve the problems. Problem solving will be monitored at all times by the teacher.
On the other hand, in the seminar sessions, the student will work, under the tutelage of the teacher, some practical situations that are related to what has been studied in the theory classes.
These sessions will also allow, students and teachers, to be aware of the evolution in the achievement of the concepts and methods that are introduced in theory classes.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Problem classes | 13 | 0.52 | 1, 4, 2, 7, 6, 3 |
Theory classes | 28 | 1.12 | 1, 2, 7, 3 |
Type: Supervised | |||
Seminars | 6 | 0.24 | 1, 4, 2, 7, 6 |
Type: Autonomous | |||
Prepare exams | 20 | 0.8 | 2, 7, 3 |
Study of the theory and resolution of problems | 65 | 2.6 | 1, 4, 2, 7, 6, 3 |
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First partial exam | (1-0.1·x)·5%(<50%) | 4 | 0.16 | 1, 2, 7, 6, 5 |
Problems and voluntary tasks | 10·x% | 6 | 0.24 | 1, 4, 2, 7, 6, 3 |
Recovery exam | 100% | 4 | 0.16 | 1, 4, 2, 7, 6, 3 |
Second partial exam | (1-0.1·x)·5%(<50%) | 4 | 0.16 | 1, 2, 7, 6 |