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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a probability vector or stochastic vector is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, and the vector gives us the
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
of that random variable, which is the standard way of characterizing a
discrete probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
..


Examples

Here are some examples of probability vectors. The vectors can be either columns or rows. * x_0=\begin0.5 \\ 0.25 \\ 0.25 \end, * x_1=\begin 0 \\ 1 \\ 0 \end, * x_2=\begin 0.65 & 0.35 \end, * x_3=\begin 0.3 & 0.5 & 0.07 & 0.1 & 0.03 \end.


Geometric interpretation

Writing out the vector components of a vector p as :p=\begin p_1 \\ p_2 \\ \vdots \\ p_n \end\quad \text \quad p=\begin p_1 & p_2 & \cdots & p_n \end the vector components must sum to one: :\sum_^n p_i = 1 Each individual component must have a probability between zero and one: :0\le p_i \le 1 for all i. Therefore, the set of stochastic vectors coincides with the standard (n-1)-simplex. It is a point if n=1, a segment if n=2, a (filled) triangle if n=3, a (filled)
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
n=4, etc.


Properties

* The mean of any probability vector is 1/n . * The shortest probability vector has the value 1/n as each component of the vector, and has a length of 1/\sqrt n. * 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. * The length of a probability vector is equal to \sqrt ; where \sigma^2 is the variance of the elements of the probability vector.


See also

*
Stochastic matrix In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
*
Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \bold ...


References

{{DEFAULTSORT:Probability Vector Probability theory Vectors (mathematics and physics)