Lecture

Gaussian Random Vectors

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

MATH-341: Linear models

Regression modelling is a fundamental tool of statistics, because it describes how the law of a random variable of interest may depend on other variables. This course aims to familiarize students with linear models and some of their extensions, which lie a ...

Description

This lecture covers Gaussian vectors, multivariate normal distribution, moment generating functions, independence of random vectors, density functions, affine transformations, isosurfaces, and coordinate distributions. It also discusses diagonal covariance matrices, chi-square and F distributions, Gaussian quadratic forms, and the Central Limit Theorem for weighted sums of random variables.

Instructors (2)

Victor Panaretos

Curriculum vitae Background & EducationPublications & PreprintsHonours & AwardsEditorial ActivitiesInvited Talks & Seminars Teaching & PhD PhD Students Ulysse Naepels, Nikitas Georgakis, Ho Yun Past EPFL PhD Students Andrea Kraus, Shahin Tavakoli, Mikael Kuusela, Yoav Zemel, Marie-Hélène Descary, Valentina Masarotto, Tomas Rubin, Tomas Masák, Laya Ghodrati, Kartik Waghmare, Leonardo Santoro Past EPFL PhD Students as codirector Matthieu Simeoni, Ophelia Miralles, Sonia Alouini, Timmy Rong Tian Tse, Krali M

Tomas Masák

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Mathematics

Algebra: Linear algebra

Related concepts (32)

Chi-squared distribution

In probability theory and statistics, the chi-squared distribution (also chi-square or -distribution) with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.

Moment-generating function

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.

Noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are the likelihood-ratio tests. Let be k independent, normally distributed random variables with means and unit variances.

Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments.

F-distribution

In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests. The F-distribution with d1 and d2 degrees of freedom is the distribution of where and are independent random variables with chi-square distributions with respective degrees of freedom and .

Related lectures (64)

Linear Models: Least Squares

Explores linear models, least squares, Gaussian vectors, and model selection methods.

Fundamental Limits of Gradient-Based Learning

Delves into the fundamental limits of gradient-based learning on neural networks, covering topics such as binomial theorem, exponential series, and moment-generating functions.

Multivariate Statistics: Normal Distribution

Introduces multivariate statistics, covering normal distribution properties and characteristic functions.

Elliptical Distributions: Properties and Applications

Covers elliptical distributions, including properties, applications, and risk management implications.

Multivariable Control