Aléatoire : une introduction aux probabilités

Ce cours d'introduction aux probabilités a la même contenu que le cours de tronc commun de première année de l'École polytechnique donné par Sylvie Méléard.

Le cours introduit graduellement la notion de variable aléatoire et culmine avec la loi des grands nombres et le théorème de la limite centrale.

Les notions mathématiques nécessaires sont introduites au fil du cours et de nombreux exercices corrigés sont proposés.

Ce cours propose aussi une introduction aux méthodes de simulations des variables aléatoires comme la méthode de Monte Carlo. Des expériences numériques interactives sont également mises à votre disposition pour vous permettre de visualiser diverses notions.

What's inside

Syllabus

L'ESPACE DE PROBABILITÉ (1/3)

L'espace de probabilité est l'objet du Cours 1 qui s'étale sur trois semaines. Après une introduction générale, cette semaine est consacrée à la notion d'expérience aléatoire et d'événement aléatoire, puis à la définition d'une probabilité sur un espace d'état fini.

L'ESPACE DE PROBABILITÉ (2/3)

Nous poursuivons le Cours 1 avec l'étude des lois de probabilités uniformes sur des espaces d'état finis. Puis nous abordons la définition générale d'une probabilité avec notamment la notion de « tribu ».

L'ESPACE DE PROBABILITÉ (3/3)

Nous achevons le Cours 1 cette semaine. Nous introduisons deux concepts centraux : le conditionnement et l'indépendance. Nous étudions ensuite un résultat très utilisé : le théorème de Borel-Cantelli. Enfin, nous introduisons un autre concept incontournable : celui de variable aléatoire.

VARIABLES ALÉATOIRES SUR UN ESPACE FINI OU DÉNOMBRABLE (1/2)

Nous commençons le Cours 2 qui dure deux semaines. Nous étudions les variables aléatoires "discrètes", c'est-à-dire des variables aléatoires prenant un nombre fini ou dénombrable de valeurs.

VARIABLES ALÉATOIRES SUR UN ESPACE FINI OU DÉNOMBRABLE (2/2)

Nous terminons le Cours 2 avec l'introduction d'un outil puissant : les fonctions génératrices. Nous étudions ensuite les couples de variables aléatoires et les variables aléatoires indépendantes.

VARIABLES ALÉATOIRES RÉELLES (1/3)

Le Cours 3, qui s'étend sur trois semaines, est consacré aux variables aléatoires qui prennent leurs valeurs dans les réels. Une nouvelle rubrique fait son apparition : « Méthodes de simulations et expériences numériques interactives ». Chaque séance du type « Illustration & expérimentation : ... » est accompagnée d'une expérience numérique interactive que vous pouvez manipuler (« A vous de jouer ! »).

VARIABLES ALÉATOIRES RÉELLES (2/3)

La notion abordée cette semaine est celle d'espérance d'un variable aléatoire.

VARIABLES ALÉATOIRES RÉELLES (3/3)

Nous terminons cette semaine le Cours 3 avec, d'une part, un résultat important permettant de calculer la loi d'une variable aléatoire et, d'autre part, des inégalités très utiles.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Explores the fundamentals of probability, aligning with the course taught at École polytechnique, a prominent institution in engineering and science

Introduces variable aléatoire—a core concept in probability—and culminates with the presentation of the law of large numbers and the central limit theorem, providing a solid foundation in statistical theory

Employs a progressive approach in introducing probability, which can be beneficial for students encountering the subject for the first time

Provides clear and detailed explanations of mathematical concepts and includes plenty of exercises with solutions to enhance comprehension

Offers interactive simulations and experiments, enabling students to visualize concepts and gain a practical understanding

Taught by Sylvie Méléard, Jean-René Chazottes, and Carl Graham, reputable instructors in the field of probability

Reviews summary

Introduction to probability part 1

This course is an introduction to probability theory. It introduces the notion of random variables and culminates with the law of large numbers and the central limit theorem. It also introduces methods for simulating random variables, such as the Monte Carlo method.

Applicable to Many Fields

"The concepts I learned in this course are applicable to many fields."

"I found the material to be very relevant to my work."

"This course gave me a strong foundation in probability theory."

Challenging but Rewarding

"This course was challenging, but I learned a lot."

"The material was difficult, but the professor was very helpful."

"I found this course to be very rewarding."

Clear and Engaging

"The professor was very clear and engaging."

"I found the lectures to be very well-organized and easy to follow."

"The professor was very knowledgeable and passionate about the subject."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Aléatoire : une introduction aux probabilités - Partie 1 with these activities:

Compile materials

Show steps

Compile all notes, assignments, reviews, and quizzes into a single resource

Show steps

Organize notes by date or topic
Categorize assignments by difficulty and skill level
Put all materials in one notebook, binder, or app

Discuss the Central Limit Theorem with a peer

Show steps

Discuss the Central Limit Theorem with a peer to better understand it

Show steps

Meet with a peer who has also taken the course
Summarize how the Central Limit Theorem works
Work together to solve practice problems

Learn about Markov Chains with an online tutorial

Show steps

Expand on the course's discussion about Markov Chains through an external online tutorial

Show steps

Find an online tutorial on Markov Chains
Follow the tutorial and take notes
Use the knowledge gained to solve practice problems

Show all three activities

Career center

Learners who complete Aléatoire : une introduction aux probabilités - Partie 1 will develop knowledge and skills that may be useful to these careers:

Risk Manager

Risk Managers use their knowledge of probability and statistics to assess and manage risk in a variety of settings, including finance, insurance, and healthcare. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and managing risk. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Risk Managers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make risk-informed decisions.

See salaries and explore the career path for Risk Manager

Quantitative Analyst

Quantitative Analysts use mathematical and statistical models to assess financial risk and make investment decisions. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with financial data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Quantitative Analysts. By taking this course, you will be well-prepared to use probability theory to solve real-world problems in the financial industry.

See salaries and explore the career path for Quantitative Analyst

Business Analyst

Business Analysts use their knowledge of probability and statistics to analyze data and make business decisions. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Business Analysts. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make data-driven business decisions.

See salaries and explore the career path for Business Analyst

Operations Research Analyst

Operations Research Analysts use their knowledge of probability and statistics to solve complex problems in a variety of industries, including manufacturing, transportation, and healthcare. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Operations Research Analysts. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make data-driven decisions.

See salaries and explore the career path for Operations Research Analyst

Statistician

Statisticians use their knowledge of probability and statistics to collect, analyze, and interpret data. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Statisticians. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make data-driven decisions.

See salaries and explore the career path for Statistician

Data Analyst

Data Analysts use their expertise in mathematics, statistics, and programming to extract meaningful insights from data. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Data Analysts. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make data-driven decisions.

See salaries and explore the career path for Data Analyst

Data Scientist

Data Scientists use their knowledge of probability and statistics to collect, analyze, and interpret data. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Data Scientists. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make data-driven decisions.

See salaries and explore the career path for Data Scientist

Machine Learning Engineer

Machine Learning Engineers use their knowledge of probability and statistics to develop and implement machine learning models. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with machine learning models. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Machine Learning Engineers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and develop machine learning models.

See salaries and explore the career path for Machine Learning Engineer

Data Engineer

Data Engineers use their knowledge of probability and statistics to design and implement data pipelines. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data pipelines. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Data Engineers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and design and implement data pipelines.

See salaries and explore the career path for Data Engineer

Epidemiologist

Epidemiologists use their knowledge of probability and statistics to study the causes and distribution of disease. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with data about disease. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Epidemiologists. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and study the causes and distribution of disease.

See salaries and explore the career path for Epidemiologist

Market Researcher

Market Researchers use their knowledge of probability and statistics to collect and analyze data about consumer behavior. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with consumer data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Market Researchers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and collect and analyze data about consumer behavior.

See salaries and explore the career path for Market Researcher

Actuary

Actuaries use their knowledge of probability and statistics to assess and manage financial risk in the insurance industry. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with insurance data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Actuaries. By taking this course, you will be well-prepared to use probability theory to solve real-world problems in the insurance industry.

See salaries and explore the career path for Actuary

Financial Analyst

Financial Analysts use their knowledge of probability and statistics to analyze financial data and make investment decisions. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with financial data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Financial Analysts. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and make investment decisions.

See salaries and explore the career path for Financial Analyst

Product Manager

Product Managers use their knowledge of probability and statistics to understand user needs and develop product roadmaps. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with user data. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Product Managers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and develop product roadmaps that meet user needs.

See salaries and explore the career path for Product Manager

Software Engineer

Software Engineers use their knowledge of probability and statistics to develop and implement software systems. This course in probability will provide you with a strong foundation in the fundamentals of probability, which is essential for understanding and working with software systems. The course covers topics such as random variables, probability distributions, and statistical inference, which are all key concepts for Software Engineers. By taking this course, you will be well-prepared to use probability theory to solve real-world problems and develop software systems.

See salaries and explore the career path for Software Engineer