Multivariate random variable

In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. More formally, a multivariate random variable is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space as each other, , where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability). Multivariate probability distribution Every random vector gives rise to a probability measure on with the Borel algebra as the underlying sigma-algebra. This measure is also known as the joint probability distribution, the joint distribution, or the multivariate distribution of the random vector. The distributions of each of the component random variables are called marginal distributions. The conditional probability distribution of given is the probability distribution of when is known to be a particular value. The cumulative distribution function of a random vector is defined as where . Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of inner products.

RANDOMIZED JOINT DIAGONALIZATION OF SYMMETRIC

Daniel Kressner, Haoze He

Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing t ...

Philadelphia2024

RANDOMIZED JOINT DIAGONALIZATION OF SYMMETRIC

Spectral Estimators for High-Dimensional Matrix Inference

Spectral Estimators for High-Dimensional Matrix Inference

RANDOMIZED JOINT DIAGONALIZATION OF SYMMETRIC