Éléments de mathématiques pour la statistique

Teacher(s)

Legrand Catherine; Uyttendaele Nathan (compensates Legrand Catherine);

Language

French

Content

This course reviews the mathematical prerequisites required for courses in the certificate and master's degree programs in statistics (general and biostatistics tracks). Specifically, it covers the core concepts of combinatorics, matrix algebra, and function analysis (including the computation of limits, derivatives, and integrals for single-variable and multivariable functions). Emphasis is placed on understanding the underlying concepts, prioritizing manual ("by-hand") calculations to achieve this.
Combinatorics

• Key definitions and main properties
  
• Solving combinatorics problems
  

Matrices and Matrix Algebra

• Basic definitions and properties
  
• Determining the dimensions, transpose, inverse, determinant, and rank of a matrix
  
• Addition, subtraction, matrix multiplication, and scalar multiplication
  
• Solving systems of linear equations using Gaussian elimination
  
• Determining the characteristic polynomial of a matrix, its eigenvalues, and corresponding eigenvectors
  

Limits, Derivatives, and Integrals

• Computing limits of single-variable functions, particularly using L'Hôpital's rule
  
• Computing derivatives of single-variable and multivariable functions
  
• Computing integrals of single-variable functions, particularly using integration by parts and substitution
  

Logarithmic and Exponential Functions

• Key properties of logarithmic and exponential functions
  
• Solving single-variable equations involving logarithmic and exponential functions
  
• Computing limits involving logarithmic and exponential functions
  
• Computing derivatives and integrals involving logarithmic and exponential functions
  

Applications of Derivatives and Integrals

• Calculating the area between two curves
  
• For single-variable functions: analyzing intervals of increase/decrease; identifying coordinates of local extrema (minima and maxima) and points with horizontal tangents
  
• For single-variable functions: analyzing concavity/convexity; identifying coordinates of inflection points
  
• Taylor series expansion of elementary functions
  
• For two-variable functions: identifying critical points where the function may reach an optimum, and classifying them as a local maximum, local minimum, inflection point, or saddle point
  
• The method of Lagrange multipliers for constrained optimization of two-variable functions
  

Teaching methods

Formal lectures and exercices sessions onsite.

Evaluation methods

A 2-hour written, closed-book exam, with the use of a basic calculator (single-line display).
The exam primarily focuses on problem-solving exercises but may also include theoretical questions.
An optional exemption test is held at the beginning of the semester, and a mandatory out-of-session evaluation will be organized at the end of the course. Both assessments are of an equivalent difficulty level to the final exam and follow a similar format. Students who score at least 14/20 on the exemption test or at least 12/20 on the out-of-session evaluation may be granted an exemption from the final exam. However, they must still register administratively for the exam regardless.
The use of artificial intelligence during these assessments is strictly prohibited.

Online resources

Moodle Website - LSTAT2011 : Eléments de mathématiques pour la statistique

Bibliography

Wackerly et al. (2002) Mathematical Statistics with Applications
Dowling E.T. (1995) Mathématiques pour l’économiste
Dodge Y. (2007) Mathématiques de base pour économistes
Lecoutre J.P. (1998) Statistique et probabilités

Teaching materials

• Slides du cours disponible sur Moodle

Faculty or entity

> LSBA