Courses LMAT1121 and LMAT1122 (real analysis/calculus, in particular bivariate integration).

*The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.*

The general aim of the course consists in giving an introduction into the thinking and the tools of probability theory and statistical analysis, with a view towards applications. The addressed topics cover the basic notions of probability (and conditional probability) and the main distributions of random vectors. The course treats the concepts of independence and correlation, and some aspects of large sample properties. For the statistical analysis, priority is given to the parametric approach (estimation of the parameters of a probability distribution) and to methods of statistical inference (hypothesis testing and confidence intervals). The statistical concepts are applied to the specific problems of analysis of variance (ANOVA) and of (simple) linear regression.

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1 | Contribution of the course to learning outcomes in the Bachelor in Mathematics programme. By the end of this activity, students will be able to: Recognise and understand a basic foundation of mathematics. Choose and use the basic tools of calculation to solve mathematical problems. Recognise the fundamental concepts of important current mathematical theories. Establish the main connections between these theories, analyse them and explain them through the use of examples. Show evidence of abstract thinking and of a critical spirit. Argue within the context of the axiomatic method. Recognise the key arguments and the structure of a proof. Distinguish between the intuition and the validity of a result and the different levels of rigorous understanding of this same result.
Learning outcomes specific to the course. The general goal of the course is to introduce the student to the notion and the tools of probability theory and statistical analysis, with a view towards applications. By the end of the course, students will be able to: Use the basic notions of probabilistic modelling, being able to worki with random variables: Apply the most frequently used techniques of probability theory (conditional probabilities and expectation, normal, Poisson and exponential laws) in various fields of application Explore structured data sets by the methods of statistical inference Apply the techniques of confidence intervals and hypothesis testing |

*The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.*

The course consists of two strongly connected parts.

First part: Probability

- Events and probabilities

- Conditional probabilities

- Independence

- Discrete random variables

- Continuous random variables

- Multivariate probability distribution (random vectors)

- Limit theorems (Central Limit Theorem, Law of large numbers)

Second part: Statistical analysis

- Random sampling and descriptive statistics

- Construction of estimators and estimation theory

- Confidence intervals

- Hypothesis testing (for means, variances and proportions)

- ANOVA

- Linear regression

The examination tests knowledge and understanding of fundamental concepts and results, ability to construct and write a coherent argument, mastery of the techniques of calculation and, above all, the applicability of the methods covered in the course to problems in the statistical analysis of data.

Site iCampus (http://icampus.uclouvain.be/claroline/course/index.php?cid=MAT$1271$\underline{ }$001$).

On the website can be found : copies of transparencies, exercice problems and their solutions, a list of formulas and statistical tables, the help file for using the statistical software, a copy of a recent exam and the detailed table of contents of the course.

D. Wackerly, W. Mendenhall, R. Scheaffer : "Mathematical Statistics with Applications" (7th ed.) 2008, Brooks/Cole.

**MATH**