2025
Probability and Statistics
Name: Probability and Statistics
Code: MAT02354L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Presentation
To provide concepts and methods of probability theory and statistical inference, looking for the interpretation and analysis of data and making statistical inference to support decision making.
Sustainable Development Goals
Learning Goals
To provide solid foundations in Probability and Statistical Inference, essential for analysing uncertainty and supporting decision-making. The course develops probabilistic and statistical reasoning, hypothesis formulation, and appropriate use of data analysis in Economics, Management, Applied Mathematics and Data Science.
By the end of the course, students should be able to:
- apply fundamental probability concepts, including independence and Bayes? theorem;
- describe discrete and continuous random variables and their distributions;
- compute and interpret moments and generating functions;
- identify and use major probability distributions;
- apply point-estimation methods (moments, MLE);
- construct confidence intervals and perform hypothesis tests;
- assess assumptions and select non-parametric alternatives;
- use R for computation, simulation, diagnostics and rigorous reporting;
- interpret results critically and recognise limitations.
By the end of the course, students should be able to:
- apply fundamental probability concepts, including independence and Bayes? theorem;
- describe discrete and continuous random variables and their distributions;
- compute and interpret moments and generating functions;
- identify and use major probability distributions;
- apply point-estimation methods (moments, MLE);
- construct confidence intervals and perform hypothesis tests;
- assess assumptions and select non-parametric alternatives;
- use R for computation, simulation, diagnostics and rigorous reporting;
- interpret results critically and recognise limitations.
Contents
The course is organised into eight modules.
1. Fundamentals of probability: random experiments, events, axioms, conditional probability, independence and Bayes? theorem.
2. Random variables: discrete and continuous; probability/density functions; joint, marginal and conditional distributions; expectation, variance, covariance and correlation.
3. Moments and generating functions: ordinary and central moments; skewness and kurtosis.
4. Main distributions: Bernoulli, Binomial, Poisson, Geometric, Negative Binomial, Uniform, Exponential, Normal, t, Chi-square and F.
5. Sampling and point estimation: sample statistics, MLE and method of moments.
6. Confidence intervals and hypothesis tests: tests for means, proportions and variances; comparison of two populations; diagnostics for normality and homoscedasticity.
7. Nonparametric tests: sign, Wilcoxon and Mann?Whitney tests.
8. Integration with R: computation, simulation, testing and interpretation.
1. Fundamentals of probability: random experiments, events, axioms, conditional probability, independence and Bayes? theorem.
2. Random variables: discrete and continuous; probability/density functions; joint, marginal and conditional distributions; expectation, variance, covariance and correlation.
3. Moments and generating functions: ordinary and central moments; skewness and kurtosis.
4. Main distributions: Bernoulli, Binomial, Poisson, Geometric, Negative Binomial, Uniform, Exponential, Normal, t, Chi-square and F.
5. Sampling and point estimation: sample statistics, MLE and method of moments.
6. Confidence intervals and hypothesis tests: tests for means, proportions and variances; comparison of two populations; diagnostics for normality and homoscedasticity.
7. Nonparametric tests: sign, Wilcoxon and Mann?Whitney tests.
8. Integration with R: computation, simulation, testing and interpretation.
Teaching Methods
The teaching and learning methodologies in this course follow an integrated, active and applied approach, combining theoretical exposition with practical exploration and digital tools, ensuring full alignment with the intended learning outcomes and syllabus.
The theoretical?practical sessions (TP) are the core of face-to-face instruction, structured to link theoretical concepts, solved examples and discussion of real cases. Each topic begins with a concise conceptual introduc¬tion, followed by applied problem-solving with real datasets from economic, business and scientific contexts. Active learning is fostered through:
? guided problem-solving in small groups and collective discussion of results;
? problem-based learning (PBL) and case studies, especially in the inference modules;
? systematic use statistical software for computation, simulation, diagnostics and graphical representation;
? critical interpretation of outputs and reflection on assumptions, errors and inferential conclusions;
? annotated examples and guided corrections.
Occasional lab sessions reinforce computational with statistical software and promote digital literacy. Students are encouraged to engage in regular autonomous work, including exercises, preparatory readings and material revision.
Autonomous work: Non-contact hours are devoted to solving exercises, preparing for tests and developing the group project. The project includes data handling, estimation and hypothesis testing, and the preparation of a technical report with critical interpretation.
Moodle serves as the organizational platform, supporting materials (slides, datasets, software scripts) and feedback throughout the semester.
These methodologies ensure: (i) diversified learning activities (theory, practice, simulation, diagnostics and communication); (ii) development of transversal competences: critical thinking, scientific communication, teamwork and academic integrity; (iii) alignment with inquiry-based and reproducible learning; (iv) use of digital tools to foster autonomy and data literacy.
The theoretical?practical sessions (TP) are the core of face-to-face instruction, structured to link theoretical concepts, solved examples and discussion of real cases. Each topic begins with a concise conceptual introduc¬tion, followed by applied problem-solving with real datasets from economic, business and scientific contexts. Active learning is fostered through:
? guided problem-solving in small groups and collective discussion of results;
? problem-based learning (PBL) and case studies, especially in the inference modules;
? systematic use statistical software for computation, simulation, diagnostics and graphical representation;
? critical interpretation of outputs and reflection on assumptions, errors and inferential conclusions;
? annotated examples and guided corrections.
Occasional lab sessions reinforce computational with statistical software and promote digital literacy. Students are encouraged to engage in regular autonomous work, including exercises, preparatory readings and material revision.
Autonomous work: Non-contact hours are devoted to solving exercises, preparing for tests and developing the group project. The project includes data handling, estimation and hypothesis testing, and the preparation of a technical report with critical interpretation.
Moodle serves as the organizational platform, supporting materials (slides, datasets, software scripts) and feedback throughout the semester.
These methodologies ensure: (i) diversified learning activities (theory, practice, simulation, diagnostics and communication); (ii) development of transversal competences: critical thinking, scientific communication, teamwork and academic integrity; (iii) alignment with inquiry-based and reproducible learning; (iv) use of digital tools to foster autonomy and data literacy.
Assessment
The evaluation will be made in accordance with paragraph 11 of article 110 of the RAUE, considering the 2 evaluation regimes foreseen: Continuous (with 2 frequencies) or Final (per Exam). The final grade (NF), for students who obtain at least 8.0 points, in each of the frequencies, will be obtained according to the following weighting NF=0.50*F1+0.50*F2, where: F1 = Grade in the 1st frequency (50%). F2 = Grade in the 2nd frequency (50%). If the NF result is greater than or equal to 9.5, even with a grade lower than 8.0 in the 2nd frequency, the classification of the normal season will be 9 values. The use of AI tools is allowed in this course as technical, analytical and learning support, as long as students understand, validate and take full responsibility for the results produced. The misuse of sources, data or results constitutes a serious violation of academic integrity. It is unacceptable to use AI in assessments or exams without authorization. Misuse will be classified as academic fraud under Article 119 of the Academic Regulations (Code of Conduct, Fraud and Plagiarism). In case of plagiarism, the test will be invalidated and participation will be made superiorly. If deemed necessary to verify the extent to which the student has developed the knowledge and skills foreseen for the curricular unit, the student may be called for an oral test, under the terms of paragraph 13 of article 110 of the Academic Regulation of the University of Évora, and the conditions provided for in paragraph 4 of article 116 and paragraph 1 of article 118 of the same Regulation are ensured.
Teaching Staff
Accreditations and Certifications
