Lecture

Central Limit Theorem: Proof

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

COM-417: Advanced probability and applications

In this course, various aspects of probability theory are considered. The first part covers the main theorems in the field (law of large numbers, central limit theorem), while the second part focuses on the theory of martingales and concentration inequalit ...

Description

This lecture presents the proof of the Central Limit Theorem using Lindeberg's principle, demonstrating the convergence of sample means to a normal distribution. The instructor explains the idea step by step, replacing variables and applying Taylor expansions.

Instructors (2)

Olivier Lévêque

Olivier Lévêque was born in Switzerland in 1971. He received the physics diploma from EPFL in 1995 and completed his PhD in mathematics at EPFL in 2001. Since then, he has been with the Laboratory of Information Theory at EPFL. He spent the academical year 2005-2006 at the Electrical Engineering Department of Stanford University, where he was appointed as lecturer. His research interests include stochastic analysis, random matrices, wireless communications and information theory.

Yanina Yurina Shkel

Teaching & PhD PhD Students Serhat Emre Coban, Anuj Kumar Yadav, Cemre Çadir Courses Advanced probability and applications COM-417 In this course, various aspects of probability theory are considered. The first part covers the main theorems in the field (law of large numbers, central limit theorem), while the second part focuses on the theory of martingales and concentration inequalities. Signal processing COM-202 Signal processing theory and applications: discrete and cont

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.

Complex normal distribution

In probability theory, the family of complex normal distributions, denoted or , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix , and the relation matrix . The standard complex normal is the univariate distribution with , , and . An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: and .

Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.

Related lectures (32)

Law of Large Numbers: Statistics

Explains the Law of Large Numbers and its application to random variables.

Probability Distributions in Environmental Studies

Explores probability distributions for random variables in air pollution and climate change studies, covering descriptive and inferential statistics.

Statistics: Laws of Large Numbers

Explores fundamental theorems in statistics, including laws of large numbers and the central limit theorem.

Acceptance-Rejection Methods: Advanced Techniques

Explores advanced techniques in acceptance-rejection methods and sampling from normal distributions.

Estimating Parameters: Confidence Intervals