Probability vector

This article treats probability vectors as elements of . For the infinite-dimensional case see stochastic sequence . ${\displaystyle \mathbb {R} ^{n}}$

A probability vector or stochastic vector is a vector with real and non-negative 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 . ${\displaystyle \mathbb {R} ^{n}}$

definition

A vector is called a probability vector or stochastic vector if for its entries ${\displaystyle v\in \mathbb {R} ^{n}}$

${\displaystyle v_{i}\geq 0}$

for everyone and ${\displaystyle i=1,\ldots ,n}$

${\displaystyle \sum _{i=1}^{n}v_{i}=1}$

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 is, for example . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle v=(0{,}2;0{,}1;0{,}7)^{\mathrm {T} }}$
• Each standard basis vector des is a probability vector. ${\displaystyle \mathbb {R} ^{n}}$
• Denotes the one vector , then is a probability vector. ${\displaystyle 1_{n}=(1,1,\dotsc ,1)^{\mathrm {T} }}$ ${\displaystyle {\tfrac {1}{n}}1_{n}}$
• 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 . ${\displaystyle X}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle (p_{1},\dotsc ,p_{n})^{\mathrm {T} }}$ ${\displaystyle p_{i}=P(X=x_{i})}$ ${\displaystyle {\tfrac {1}{n}}1_{n}}$

properties

• If there is a column stochastic matrix and a probability vector, then it is again a stochastic vector. ${\displaystyle A}$ ${\displaystyle v}$ ${\displaystyle Av}$
• The set of probability vectors of length is closed and convex ; so it is a polyhedron in -dimensional space, namely the convex hull of the standard basis vectors. ${\displaystyle n}$ ${\displaystyle n}$
• For each probability vector is the sum norm . ${\displaystyle \Vert v\Vert _{1}=1}$

use

In stochastics, probability vectors are used to describe the probability of a system being in certain states. If the system has different states, 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 . ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle \pi }$

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 : Lineare Algebra . Basics and applications (=  Springer textbook ). Springer, Berlin 2012, ISBN 978-3-642-32185-6 .