Probability vector
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 .
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 is, for example .
- Each 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 ; so 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 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 .
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 .