Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable

Multivariate random variable

In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.More formally, a multivariate random...

.
In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a probability vector or stochastic vector is a vector

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 with non-negative entries that add up to one.

The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function

Probability mass function

In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

 of that random variable, which is the standard way of characterizing a discrete probability distribution.

Here are some examples of probability vectors:



Writing out the vector components of a vector as


the vector components must sum to one:


One also has the requirement that each individual component must have a probability between zero and one:


for all . These two requirements show that stochastic vectors have a geometric interpretation: A stochastic vector is a point on the "far face" of a standard orthogonal simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

. That is, a stochastic vector uniquely identifies a point on the face opposite of the orthogonal corner of the standard simplex.

Some Properties of dimensional Probability Vectors

Probability vectors of dimension are contained within an dimensional unit hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

.

The mean of a probability vector is .

The shortest probability vector has the value as each component of the vector, and has a length of .

The longest probability vector has the value 1 in a single component and 0 in all others, and has a length of 1.

The shortest vector corresponds to maximum uncertainty, the longest to maximum certainty.

No two probability vectors in the dimensional unit hypersphere are collinear unless they are identical.

The length of a probability vector is equal to ; where is the variance of the elements of the probability vector.