Probability

Definition of Provability space, examples. 

Random variables, expected value, variance and covariance, linear correlation. Law of a random variable. Examples. Cumulative distribution function. 

Examples of cdf. Absolutely continuous random variables, density function. Examples. Standard random variables: exponentials, gaussian, Gamma, Cauchy, log-normal. Change of density.

Exercises on random variables, cdfs, densities. Multivariate random variable, mean and covariance matrix.

Multivariate random variables: law, cdf, absolutely continuous r.v. and densities. Multivariate Gaussian. 

Exercises on multivariate distributions. Mapping multivariate random variables, exercise. 

Exercises on multivariate random variables. Fourier Transform of a Borel measure. FT of a multivariate Gaussian distribution. 

Characteristic function of a random variable, examples and main properties. Uniqueness theorem. Exercises.

(no video sorry)

Independent events and random variables. Characterization of independence through joint cdf and densities. Exercise. 

Exercises on independence. Kac theorem on characteristic function of independent random variables. 

Exercises on conditional expectation

Convergence in distribution. Exercises on convergence of random variables. 

Exercises on convergence of sequences of random variables. 

Weak law of large numbers: Chebishev bound, applications to monte-Carlo method and Bernstein-Weierstrass polynomial approximation. 

Exercises on W/SLLN. Central limit theorem. 

Exercise CLT. Definition of Brownian motion. The Levy-Ciesielskii construction. 

Exercises on Brownian motion. BM paths regularity. 

Exam simulation exercises