This course introduces the basic concepts of probability theory, i.e. the mathematical analysis of phenomena in which chance is involved. Particular emphasis will be placed on the two major concepts that underpin this theory: conditioning and the law of large numbers. The aim of the course is to develop probabilistic reasoning, probabilistic modeling and simulation. Probabilistic modeling is fundamental to a wide range of applications. The course is illustrated by examples and numerical experiments on Python notebooks. It also introduces a few notions of measurement theory (which is the axiomatic foundation of probability theory) and provides an introduction to statistics. During this teaching, students will carry out a Python simulation project in pairs, which will count towards the module grade.
Contents :
Lesson 1: Examples of discrete models
- discrete probability, uniform law and combinatorial calculus, fundamental set, events, sigma-field
- conditional probability, total probability, Bayes formula
- independent events
- Borel-Cantelli lemma
- random variables in a countable space at most, discrete distributions, usual distributions, expectation, generating functions
- conditional distributions and expectations for discrete laws
Lesson 2: Measurement, probability and real random variables
- Sigma-field, Borelian sigma-field, probability (abstract measure), probability space
- Lebesgue measure
- real random variable (measurable function), distribution
- distribution function
- real density random variable (integration and derivation)
- uniform, exponential and normal distributions
- simulation by inversion of the distribution function
Lesson 3: Expectations of real random variables
- expectation (Lebesgue integral for abstract measure), variance, transfer property
- expectations and variances of usual distributions
- calculating the law of a real a.v. using the dumb function method (change of variable)
- covariance between two random variables, correlation coefficient
- inequalities: Markov, Jensen, Bienaymé-Chebyshev, Cauchy-Schwarz
- random vector
Lesson 4: Random vectors
- characterization, distribution with density, expectation
- Fubini's theorem
- covariance and linear regression
- conditional distributions and conditional expectation - independent random vectors
- rejection simulation method
Lesson 5: Random vectors: calculation of laws, Gaussian vectors
- sum of independent random variables
- multidimensional change of variables
- Gaussian distribution simulation by Box-Muller algorithm
- Gaussian vector and properties
- linear regression
Lesson 6: Convergence theorems
- different modes of convergence and their relationships
- almost sure convergence
- convergence in mean
- convergence in probability
- monotone convergence and dominated convergence theorems
- laws of large numbers
Lesson 7: Convergence in law - Central limit theorem
- convergence in law
- ... and convergence in probability
- slutsky's theorem
- characteristic functions
- characterization of a Gaussian vector
- Lévy's theorem
- central limit theorem
Lesson 8: Applications of the central limit theorem: Statistical estimation
- Delta method
- central limit theorem for random vectors
- statistics:
- estimators
- bias, mean-squared risk
- convergence, asymptotic normality
- method of moments
- maximum likelihood estimator
Lesson 9: Statistics: Confidence intervals
- Confidence intervals
- Exact calculation in the Gaussian case
- Approximate calculation in the general case
Course language : French
ECTS credits : 5
