#### Objective

#### Content

- Probability Law. Probability Space.
- Random variables. First and second Borel-Cantelli lemma. Distribution function of a random variable.
- Expectation on a probability space. Inequalities (Tchebychev, Markov, Schwartz and Jensen).
- Conditional expectation. Joint distribution function and joint probability density function of two random variables. Independence. Covariance and properties. Correlation coefficient. Conditional expectation and conditional variance : properties. Calculating probability by conditionning.
- Moment generating function (m.g.f.) and characteristic function (c.f.). Inversion theorem.
- Random vectors and Gaussian vectors : marginal density, conditional probability density function, variance-covariance matrix, m.g.f. of a random vector, transformation (Jacobian), m.g.f. of a multivariate normal distribution . Case of two dimensions.
- Convergence of random variables: almost sure convergence, convergence in probability and in distribution. Weak law of large numbers and strong law of large numbers. Central limit theorem.
- Stochastic process. Chain of Markov. Chapman-Kolmogorov Equations. Doob decomposition.