A multivariate random variable or random vector is a vector X = (X₁, ..., X_n) whose components are scalar-valued random variables on the same probability space (Ω, P). Every such random vector gives rise to a probability measure on Rⁿ with the Borel algebra as underlying sigma-algebra. This measure is also known as the joint distribution of the random vector. The distributions of each of the component random variables X_i are called marginal distributions. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, a probability space or probability measure is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...