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Probabilities Description

Key Elements

Code

MATH 404

Formation

M1 Mathematics

Semester

1

Credits

6

Number of Teaching Hours

36

Number of Tutoring Sessions

24

Number of Laboratory Sessions

0

Content

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.