Differential Calculus (1st year - 1st term) Integral Calculus (1st year - 2nd term) Vector Calculus (1st year - 2nd term) Integration and Measure (2nd year - 1st term)

1. Knowing the theoretical foundations and calculus rules of Probability Theory. 2. Resolution of problems of Probabilistic Nature.

K04: now the principles of probability calculus and statistical inference and how to apply them in solving real-life problems S03: Apply mathematical language and abstract-rigorous reasoning in the enunciation and demonstration of results in various areas of mathematics. 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. Probability and random phenomena 1.1 Random phenomena, sample space, events 1.2 Definition of probability and elementary properties 1.3 Conditional probability and independence 1.4 Total probability rule and Bayes' formula 2. Random variables 2.1 Definition of random variable 2.2 Distribution of a random variable 2.3 Expectation and other characteristic features of a random variable 2.4 Transformations of random variables 3. Common distribution models 3.1 Discrete probability models 3.1.1 Binomial distribution 3.1.2 Geometric distribution 3.1.3 Poisson distribution 3.2 Continuous probability models 3.2.1 Uniform distribution 3.2.1 Exponential distribution 3.2.3 Normal distribution 4. Jointly distributed random variables 4.1 Definition of random vector, joint, marginal, and conditional distributions 4.2 Independent random variables 4.3 Some multivariate distribution models 4.4 Transformations of random vectors 5. Properties of the expectation 5.1 Expectations of transformation of random variables 5.2 Covariance, variance of sums, and correlation 5.3 Conditional expectation, law of iterlated expectation 5.4 Moment generating functions 6. Limit Theorems 6.1 Chebyshev's inequality 6.2 Convergence in probability, the Weak Law of Large Numbers 6.3 Convergence in distribution, the Central Limit Theorem 6.4 Almost sure convergence, the Strong Law of Large Numbers

THEORETICAL-PRACTICAL CLASSES. [44 hours with 100% classroom instruction, 1.76 ECTS] Knowledge and concepts students must acquire. Student receive course notes and will have basic reference texts to facilitate following the classes and carrying out follow up work. Students partake in exercises to resolve practical problems and participate in workshops and evaluation tests, all geared towards acquiring the necessary capabilities. TUTORING SESSIONS. [4 hours of tutoring with 100% on-site attendance, 0.16 ECTS] Individualized attendance (individual tutoring) or in-group (group tutoring) for students with a teacher. STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.92 ECTS] FINAL EXAM. [4 hours with 100% on site, 0.16 ECTS] Global assessment of knowledge, skills and capacities acquired throughout the course. METHODOLOGIES THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual support in which the subject's main concepts are developed, while providing material and bibliography to complement student learning. PRACTICAL CLASS. Resolution of practical cases and problem, posed by the teacher, and carried out individually or in a group. TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or in-group (group tutoring sessions) for students with a teacher as tutor. LABORATORY PRACTICAL SESSIONS. Applied/experimental learning/teaching in workshops and laboratories under the tutor's supervision.