In this course, the Probability Theory is presented in the context of Measurement Theory and the principles of the Stochastic Processes Theory. In this way, the basic training acquired by the student in the second year of the degree with the Probabilty Calculus course is completed by carrying out a solid and systematic development of the principles, results and applications of the Probability Theory.





This course, together with the subjects Mathematical Programming and Multivariate Analysis, form the Module M14 of the Degree in Mathematics which is called Extension of Statistics and Operations Research. The objective of this module is to provide knowledge and techniques of probability, statistics and operations research so that the student acquires a basic and horizontal training of these courses that allows them to understand and apply such knowledge and skills in multiple interrelated directions. These three subjects can be followed independently.





The following subjects that are taken in the first, second and third year of the degree are desirable requirements when taking this subject: Calculation of probabilities, Measurement and integration, Complex analysis and Differential and Integral Calculus I and II.

COMPETENCIES:

M14CM01- Deep knowledge of the concepts and results of probability calculation, statistics and mathematical programming.

M14CM03- Correct use of terminology related to random phenomena, data analysis and optimization of linear functions.

M14CM04- Correct modeling of common situations related to random phenomena and data processing.

M14CM06- Correct selection of the appropriate analysis technique, according to the goal achieved in the analysis of these situations.

M14CM07- Correct calculations or graphic displays, when required by such situations, using appropriate theoretical and/or computational resources.

M14CM08- Critically interpret the results of the analysis carried out.



LEARNING RESULTS:

To know how to formulate, solve and interpret calculation problems of probabilities and stochastic processes.

1. PROBABILITY SPACES: probability and measure, probability spaces, conditional probability,

independence of events and collection of events.

2. RANDOM VARIABLES: measurable functions, probability distribution, independence of random variables.

3. EXPECTATION: expectation as integral, properties, moments, main inequalities.

4. CHARACTERISTIC FUNCTIONS: concept and main properties, derivatives and moments, inverse functions,

identification of characteristic functions.

5. CONVERGENCE: modes of convergence of random variables, mutual relations, strong and weak laws of large numbers, convergence of random series, the central limit theorem and its generalizations.

6. CONDITIONAL EXPECTATION: concept and main properties, martingales, convergence of martingales.

7. STOCHASTIC PROCESSES: Markov chains, other stochastic processes, fundamentals of process theory.

In theory classes, basic theoretical concepts and results are explained, developed and illustrated. The problem classes show the practical aspects of the theory presented in the lectures. They can also be used to assign tasks to be done, to show instructions for doing them or to explain certain tasks. In the seminars, the student will take a more active role and will have to demonstrate the skills acquired up to that point in the skills studied. Depending on the session, different activities will be carried out, such as the theoretical and/or practical tasks assigned to them will be presented, individual or group work will be done, problem solving,...

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • See orientations (%): 100

CONTINUOUS EVALUATION GUIDELINES:

The evaluation of the subject will consists of presentations of problem-solving works, as well as some written tests. Exactly:

Partial writen exam: %25.

Solving problems in class, delivering and presenting proposed problems and/or theoretical works, participating in seminars and tutorials: % 20.

Final writen examen: % 55



The partial written test and the final written test are compulsory.



The 20% assessment of problem solving in class, delivery and presentation of proposed problems and/or theory assignments, participation in seminars and tutorials will be optional, always taking into account that, if continuous assessment has been chosen, the non-delivery / realization / presentation will imply the automatic loss of this percentage in the note.



The student who does not appear for the final written test that is carried out on the date of the ordinary call will be evaluated as "Not presented".



The student who does not want to participate in the continuous evaluation may officially renounce it by means of a letter addressed to the responsible teacher that must be delivered within a maximum period of 15 weeks from the beginning of the semester.



GUIDELINES FOR THE FINAL EVALUATION:

A written exam will be carried out on the date of the ordinary call whose qualification will be 100% of the note.



CONSIDERATIONS TO TAKE INTO ACCOUNT:

When evaluating, the following will be taken into account:

In the written tests: the precision and rigor in the definitions, properties and reasoning, the correctness of the results and developments, the correct use of mathematical language and the correct method of reasoning (clear, orderly and reasoned explanations of the steps followed and arguments used).

In the presentations and delivery of works: the precision and rigor in the definitions, properties and reasoning, the correction in the results and in the developments, the adequate use of mathematical language both in written and oral form and the clear, orderly and reasoned justifications of the arguments used.

There will be a written exam and the obtained mark will be 100% of the note.

Lists of material and problems taught in class and available in the subject's virtual classroom.

Basic bibliography

G.R. GRIMMETT, D.R. STIRZAKER, Probability and Random processes, Oxford Science Publications, 1992

A.F. KARR, Probability, Springer Verlag, 1993.

S.I. RESNICK, A Probability Path, Birkhäuser, 1999.

In-depth bibliography

P. BILLINGSLEY, Probability and Measure, Wiley, New York, 1986.
J. NEVEU, Martingales a temps discret, Dunod, 1972.
A. N. SHIRYAYEV, Probability, Springer-Verlag, New York, 1996.

Web addresses

Virtual classroom to support face-to-face teaching: https://egela.ehu.eus/
Probability Web: https://www.stat.berkeley.edu/~jpopen/probweb/