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Lecture

Joint Distribution of Gaussian Random Vectors

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 discusses the joint distribution of Gaussian random vectors, focusing on when Gaussian random vectors admit a joint PDF. The key criterion for a Gaussian random vector to have a joint PDF is that its covariance matrix must be positive definite. The lecture covers the case of a centered Gaussian vector with a diagonal covariance matrix, providing the general expression for the joint PDF. It then extends the discussion to Gaussian random vectors with a positive definite covariance matrix, emphasizing the importance of the spectral theorem in proving the existence of a joint PDF. The lecture concludes by showing how any centered Gaussian random vector can be expressed as a product of two random vectors, facilitating computations for correlated Gaussian random vectors.

Instructors (2)

Yanina Yurina Shkel

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.

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (21)

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.

Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions.

Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative.

Estimation of covariance matrices

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix.

Sample mean and covariance

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger population of numbers, where "population" indicates not number of people but the entirety of relevant data, whether collected or not. A sample of 40 companies' sales from the Fortune 500 might be used for convenience instead of looking at the population, all 500 companies' sales.

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