Currículo

Probability and Statistics PRBEST-E

Contextos

ECTS

6.0 (para cálculo da média)

Objectivos

1. To acquire the basic notions of Probability which are needed for uncertainty quantification and for reasoning in a context of uncertainty. 2. To apprehend the concept of random variable and to know the probability distributions that are commonly used in basic modeling of observable phenomena. 3. To understand the formalization of the problem of statistical inference to make inference about unknown population quantities from a sample. 4. To know the basic tools of statistical inference, including point estimation, interval estimation and hypotheses testing, particularly in the case of Gaussian populations and in the case of large sample sizes. 5. To be aware of the limitations and the difficulties associated with the problem of statistical inference.

Programa

1. Probability: Kolmogorov axioms, conditional probability, independent events. 2. Random variables: cumulative distribution function, discrete and continuous random variables. 3. Expected values and parameters: mean value, variance, quantiles, median. 4. Bivariate random variables: joint distribution and marginal distributions. Independence. Covariance. 5. Important distributions: binomial distribution and Poisson distribution. Continuous uniform distribution. Exponential distribution. Gaussian distribution. 6. Probability versus statistics. The concept of random sample and the concept of statistic. Properties of the sample mean and of the sample variance. 7. Sampling distribution of the sample mean and of the sample variance in the context of a Gaussian population. Chi-square and Student-t distributions. Large samples: central limit theorem. 8. Point estimation. Properties of estimators. The maximum likelihood method and the method of moments. 9. Interval estimation: basic ideas. Pivotal quantity. The construction of confidence intervals in the context of a Gaussian distribution. The case of large sample sizes. 10. Hypotheses testing: basic ideas. Type I and type II errors in the context of simple hypotheses. Tests for composite hypotheses. P-value. Hypotheses testing in the context of a Gaussian population. Large sample sizes.

Método de Avaliação

The contact with the students is divided between lectures (two per week, 90 minutes each) and labs (one per week, 120 minutes long). In the lectures, we introduce the concepts and the main results, along with suggestions on how to study the material. Plenty of examples complement the presentation of the ideas. In the labs, the instructor will clarify any questions that the students might have, and selected exercises will be solved. These exercises will be taken from the recommended manual (listed in the recommended bibliography) or adapted from previous exams. The set of exercises to be covered in each lab will be announced in the lectures. The evaluation of the students will be based on written exams, according to ISEG's RGAC. If there is a week during the semester dedicated to evaluation, we may consider having a test during that week, which will be worth 50% of the final grade. For the regular exam date, students may choose to be evaluated on the remainder of the syllabus or to take an exam covering the whole syllabus. On the resit date, there will only be an exam.

Carga Horária

Bibliografia

Não foi definida bibliografia principal

Disciplinas de Execução

2025/2026 - 1 Semestre