**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Central Limit Theorem: Illustration and Applications

Description

This lecture covers the Central Limit Theorem, explaining how the behavior of averages of independent and identically distributed random variables changes as the sample size increases. It discusses the estimation of probabilities using normal distributions and the approximation of sums of independent random variables. The lecture also explores the concept of empirical quantiles and their applications in statistical analysis.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

MATH-233: Probability and statistics

The course gives an introduction to probability and statistics for physicists.

Instructors (2)

Related concepts (75)

Related lectures (55)

Half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let follow an ordinary normal distribution, . Then, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Using the parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by where .

Quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups). The groups created are termed halves, thirds, quarters, etc.

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 .

Exchangeable random variables

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models.

Complex random variable

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.

Periodic Orbits of Hamiltonian Systems

Covers the theory of periodic orbits of Hamiltonian systems and the Conley-Zehnder index.

Linear Algebra: Matrix OperationsMATH-212: Analyse numérique et optimisation

Covers matrix operations and properties, including eigenvalues and eigenvectors.

Modes of Convergence of Random VariablesMATH-232: Probability and statistics

Covers the modes of convergence of random variables and the Central Limit Theorem, discussing implications and approximations.

Holography in Classical GravityPHYS-739: Conformal Field theory and Gravity

Covers the concept of holography in classical gravity and its relation to string theory.

Eigenstate Thermalization Hypothesis

Explores the Eigenstate Thermalization Hypothesis in quantum systems, emphasizing the random matrix theory and the behavior of observables in thermal equilibrium.