ECTS credits ECTS credits: 5
ECTS Hours Rules/Memories Student's work ECTS: 85 Hours of tutorials: 5 Expository Class: 15 Interactive Classroom: 20 Total: 125
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Statistics and Operations Research
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
The objective of the subject is for the student to master the fundamental aspects of probability theory and to be able to apply these concepts in other disciplines such as mathematical statistics.
Unit 1. Introduction
Unit 2. Probability space
Definition of probability space. Construction of probability spaces.
Unit 3. Foundations of probability theory.
Random variable. Independence. Borel-Cantelli lemmas. Kolmogorov's law zero-one.
Unit 4. Expected value.
Definition. Properties.
Unit 5. Distribution of a random variable.
Definition. Variable change theorem. Distribution examples.
Unit 6. Results of probability theory.
Limits and expectation. Derivatives and expectation. Moment generating function. Fubini's theorem.
Unit 7. Inequalities and convergence.
Inequalities. Convergence of random variables. Strong and weak law of large numbers.
Unit 8. Weak convergence.
Definition. Properties. Relationship with other modes of convergence.
Unit 9. Characteristic function.
Definition. Properties. Inversion theorem, uniqueness and continuity.
Unit 9. Central Limit Theorem.
Basic
Athreya, K. y Lahiri, S. (2006), Measure Theory and Probability Theory, Springer. Billingsley, P. (1995), Probability and Measure, Wiley.
Chow, Y. S. y Teicher, H. (1997) Probability Theory: Independence, Interchangeability, Martingales, Springer
Durret, R (2004), Probability: Theory and Examples. Duxbury Press.
Rosenthal, J. S. (2006), A first look at rigorous probability theory, World Scientific Publishing Co.
Complementary
Apostol, T. (1974), Mathematical Analysis, Adison Wesley. Royden, H. L. (1988), Real Analysis, Macmillan Publishing Co.
In this matter the basic, general and transversal competences included in the memory of the title will be worked on. The specific competences that will be promoted in this area are indicated below:
Specific competences:
E1 - Know, identify, model, study and solve complex statistical and operational research problems, in a scientific, technological or professional context, arising from real applications.
E3 - Acquire advanced knowledge of the theoretical foundations underlying the different methodologies of statistics and operational research, which allow their specialized professional development.
E4 - Acquire the necessary skills in the theoretical-practical management of probability theory and random variables that allow their professional development in the scientific / academic, technological or specialized and multidisciplinary professional field.
E5 - To deepen the knowledge in the specialized theoretical-practical foundations of modeling and study of different types of dependency relationships between statistical variables.
E6 - Acquire advanced theoretical-practical knowledge of different mathematical techniques, specifically oriented to aid in decision-making, and develop reflective capacity to evaluate and decide between different perspectives in complex contexts.
E8 - Acquire advanced theoretical-practical knowledge of techniques for making inferences and contrasts related to variables and parameters of a statistical model, and know how to apply them with sufficient autonomy in a scientific, technological or professional context.
The activity at the classroom of the students will be a maximum of 35 hours between expository and interactive teaching. In the expository part, the teaching staff will make use of multimedia presentations, while in the interactive part, the students will solve different questions raised about the contents of the subject.
The students will have, through the virtual campus of the subject, the teaching material (presentations, notes, exercises) of the subject. Throughout the course, works will be proposed that students must solve with the tutor's supervision. This tutoring will be done in small groups
At the first opportunity, it will be evaluated through continuous evaluation and a final test. Continuous evaluation will consist of the delivery and review of different works proposed throughout the course. The exercises will be of various levels of theoretical / practical difficulty.
Thus, the most advanced will allow evaluating the acquisition of skills CB6, CB7, CG4, CT1, E3 and E4.
More applied exercises will be presented that will allow modeling complex situations, developing skills CB8, CG1, CG5, CT2, E1, E5, E6.
Autonomy will be valued in the resolution of proposals, as specified in competencies CB10, E8.
A group work will be proposed that will be presented orally and that will allow evaluating the competences CB9, CG2, CG3, CT4, CT5.
The final test will evaluate the E3 and E4 competencies.
In the first opportunity the weight of the continuous evaluation will be 70%, the remaining 30% corresponding to the final exam.
In the second opportunity, the continuous assessment grade will be complemented with a theoretical / practical exam. The final grade will be the weighted average of the continuous evaluation of the first part of the subject and the final exam. The weights will be 40% and 60% respectively.
It is considered that the personal work time of the students to pass the subject is 125 hours distributed as follows:
1) On-site activity (35):
2) Study of the material (35): It is estimated 1 hour for each hour of activity
3) Continuous assessment work (55 hours)
To successfully pass the subject, it is advisable to attend the expository and interactive teaching sessions, with daily monitoring of the work carried out in the classroom being essential. Likewise, it is recommended that the student have prior knowledge of probability calculation, and a good command of abstract mathematical concepts.
The development of the contents of the subject will be carried out taking into account that the competences to be acquired by the students must meet the MECES3 level. The basic concepts of probability theory will be presented and studied in depth, from a mathematical perspective, highlighting its instrumental application or as theoretical support for different inferential techniques.
In cases of fraudulent performance of exercises or tests, the provisions of the respective regulations of the universities participating in the Master in Statistical Techniques will apply.
Alberto Rodriguez Casal
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Statistics and Operations Research
- alberto.rodriguez.casal [at] usc.es
- Category
- Professor: University Professor
| 01.08.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 04 |
| 06.19.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 04 |