What is a probability vector

Probability vector

A Probability vector or stochastic vector is a vector with real and nonnegative entries, the sum of which is one. Probability vectors are used in both linear algebra and stochastics. Probability vectors should not be confused with random vectors, these are random variables with values ​​in.

definition

A vector is called a probability vector or stochastic vector if for its entries

for everyone and

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

Examples

  • A probability vector des is, for example.
  • Every standard basis vector des is a probability vector.
  • Denotes the one vector, then is a probability vector.
  • In general, the following applies: If a random variable that only takes on a finite number of values, then the probabilities are a probability vector. In this way, for example, represents a discrete uniform distribution.

properties

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

use

In stochastics, probability vectors are used to describe the probability of a system being in certain states. If the system has different states, then the -th component of a probability vector is precisely the probability that the system is in the state. In stochastics, in contrast to linear algebra, probability vectors are often defined as line vectors and are usually denoted by the symbol.

They are also used to define stochastic matrices. In the case of a row stochastic matrix, the row vectors are stochastic; in the case of a column stochastic matrix, the column vectors are correspondingly. A matrix in which both row and column vectors are probability vectors is called a double-stochastic matrix.

literature

  • Peter Knabner, Wolf Barth: Linear Algebra. Basics and Applications (=Springer textbook). 1st edition. Springer, Berlin 2012, ISBN 978-3-642-32185-6 (996 pages).