Prerequisites in Analysis
Sets: Sets, subset and inclusion. Union and intersection of sets, difference of two sets, complement of a subset. Countable sets. Mapping between two sets. Direct and inverse image of a set by a mapping. Inverse mapping. Injective, surjective and bijective mappings. Composition of two mappings.
Sequences and series of real numbers: Limit of real valued sequences, convergence and divergence. Equivalent sequences. Monotone sequences. Partial sums and convergence of series. Absolute convergence of a series. Geometric and exponential series.
Functions defined on the real line: Definition of a function, graph of a function. Limit of a function at a given point and continuity. Derivative of a function, tangent equation and usual properties of derivatives (derivation of a product, a quotient or a composition of functions, derivative of the inverse). Odd, even and periodic functions. Derivative and variations of a function. Usual functions: exponential, logarithm, power, sine and cosine. Convex functions. Primitive of a function. Riemann sums, Riemann integral and area under a curve. Basic properties of the Riemann integral. Integration by parts and change of variables formula.
Functions of several variables: Partial derivatives. Gradient. Hessian matrix. Taylor expansions. Optimization without constraints: first-order and second-order conditions. The case of convex functions. Integration of functions of several variables. Fubini theorem.
Norm, scalar product and Euclidean space: Definition of a norm, of a distance. Scalar product and associated norm. Cauchy-Schwarz inequality. Orthogonal vectors. Euclidean spaces. Orthonormal basis. Gram-Schmidt process. Distance between a point and a vector subspace, orthogonal projection and expression in an orthonormal basis. Orthogonal complement of a vector subspace. Affine hyperplanes in Euclidean spaces
Prerequisites in Algebra
Complex numbers: Real and imaginary part of a complex number. Modulus and trigonometric form. Elementary algebraic operations. Exponential of a complex number. Quadratic equations.
Polynomials: Roots of a polynomial, divisibility. Polynomial functions. Degree of a polynomial and roots multiplicity.
Vector spaces: Notion of vector space and vector subspace. Vector subspace generated by a family of vectors. The vector space ℝn. Linearly independent vectors. Basis. Coordinates of a vector in a basis. Sum of vector subspaces. Complements of a vector subspace. Vector spaces of finite dimension. Dimension of a vector subspace, rank of a system of vectors. Linear maps. Kernel and image of a linear map. Rank-nullity theorem. Linear form and hyperplane. Affine subspaces of a vector space.
Matrices: Sum and product of matrices. Transpose of a matrix. Inverse matrix. Rank of a matrix. Trace. Link between matrices and linear maps. Link between matrices and linear
systems. Determinant of matrices and properties.
Spectral decomposition of a square matrix: Eigenvalues, eigenvectors. Basis of eigenvectors. Diagonalization of a matrix. Spectral decomposition of symmetric matrices, positive semi-definite matrices and orthogonal projection matrices.
Prerequisites in Probability & Statistics
Combinatorics: Cardinality of a set. Factorial of an integer, binomial coefficients. Binomial expansion.
Probability space: Random experiments, events and probability measures. Basic properties of a probability measure. Conditional probabilities. Bayes formula. Formula of total probability. Independent events.
Random variables: Discrete or continuous random variables. Usual discrete probability distributions (Bernoulli, Poisson, uniform, exponential, normal). Independence of random variables. Expectation of a random variable. Variance and covariance. Conditional distribution and conditional expectation.
Convergence and limit theorems: convergence in probability, convergence in law, law of large numbers, central limit theorem.
Exploratory Statistics and Data analysis: Mean, median, mode, range, standard deviation, interquartile range, quartiles and percentiles; interpretation of data in tables and graphs (line graphs, bar graphs, circle graphs, boxplots, scatterplots and frequency distributions); principal components analysis.
Statistical inference: statistical model, likelihood, method of moments, point estimation, bias, mean square error, confidence intervals, tests, p-value, chi-squared tests, Student’s t-test, Kolmogorov-Smirnov test.
Regression analysis and Time series: linear regression, analysis of variance, logistic regression, regression trees, autoregressive linear process, moving averages.
Prerequisites in Programming, Algorithms & Data Structures
Abstract types: setting arrays, trees, dictionaries.
Classic algorithmic patterns: greedy approach, divide and conquer, dynamic programming.
Programming: Writing and compiling a program, debugging programs, input/output using files, use of variables, control structures and loops, exception handling, writing functions, methods, procedures.
Languages and statistical software: intermediate level in Python or R
Relational database management system: SQL language, variable type, create a table, update a table, SELECT/DELETE/INSERT INTO queries, SQL scripts.