This is a refresher course in the main probabilistic tools for students arriving directly in the second year. These tools will be useful for making the most of courses in mathematical statistics and derived disciplines: econometrics, models with qualitative variables, sample survey theory, temporal series…
The mathematical formalisation, based on the theory of integration, is covered in a fairly brief preliminary chapter restating the main results and notations. Thereafter the emphasis is placed on calculating laws and changing variables, due to their practical importance, and on specific concepts in probability theory which give rise to relatively detailed developments: conditional expected values and convergences.
- Probabilities, measures and integration: elements for an epistemological approach.
- Random variables and laws– Studies of moments and the geometric properties of variance-covariance matrices, determining and characterising laws, changes of variables, examples of common laws.
- Independence, conditioning and conditional expected values –Geometrical theory in Hilbert space $L^2$ and general theory for variables in $L^1$, general theory of conditional probabilities with application to the calculation of conditional expected values.
- Study of normal random variables –Derived laws, studies of linear and quadratic transformations of normal laws.
- General convergence problem –Links between different types of convergence, sequences and series of independent random variables, limit theorems in asymptotic theory.