Probability vector

A probability vector or stochastic vector is a vector with real and non-negative entries whose sum to one. Probability vectors are used both in linear algebra and in the stochastics.

Definition

A vector called probability vector or stochastic vector when its entries for

For all and

Applies. In a probability vector, therefore, are all the entries greater than or equal to zero and the sum of the entries to one.

Examples

• A probability vector is the example.
• Each standard basis vector of a probability vector.
• Refers to one vector, then is a probability vector.
• As a general rule: If a random variable that takes only finitely many values ​​, then the probabilities a probability vector. For example, represented in this way is a discrete uniform distribution.

Properties

• Spaltenstochastische a matrix and a probability vector, it is a stochastic vector again.
• The set of probability vectors of length is closed and convex; Thus, it is a polyhedron in -dimensional space, namely, the convex hull of the standard basis vectors.
• For each probability vector is the sum norm.

Use

In the stochastic probability vectors are used to describe the probability of a system in certain states. The system has various states, the th component of a likelihood vector is just the probability that the system is in the state. In the stochastic probability vectors, in contrast to linear algebra often defined as row vectors and usually denoted by the symbol.

Furthermore, they are also used for the definition of stochastic matrices. In a zeilenstochastischen matrix, the row vectors are stochastically, at a spaltenstochastischen matrix corresponding to the column vectors. A matrix in which both row and column vectors are probability vectors is called a double stochastic matrix.