Probability vector: Difference between revisions

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{{Short description|Vector with non-negative entries that add up to one}}

''[[{{redirect|Stochastic vector]] redirects here. For |the concept of a random vector, see [[|Multivariate random variable]].''}}

{{Unreferenced|date=December 2009}}

In [[mathematics]] and [[statistics]], a '''probability vector''' or '''stochastic vector''' is a [[vector space|vector]] 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]], and the vector gives us the [[probability mass function]] of that random variable, which is the standard way of characterizing a [[discrete probability distribution]].<ref>{{citation

| last = Jacobs | first = Konrad

| doi = 10.1007/978-3-0348-8645-1

| isbn = 3-7643-2591-7

| mr = 1139766

| page = 45

| publisher = Birkhäuser Verlag, Basel

| series = Basler Lehrbücher [Basel Textbooks]

| title = Discrete Stochastics

| url = https://books.google.com/books?id=2Rv_i4-01JEC&pg=PA45

| volume = 3

| year = 1992}}.</ref>

==Examples==

Here are some examples of probability vectors:. The vectors can be either columns or rows.

*<math>

x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix},\;</math>

*<math>

x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},\;</math>

*<math>

x_2=\begin{bmatrix} 0.65 \\& 0.35 \end{bmatrix},\;</math>

*<math>

x_3=\begin{bmatrix} 0.3 \\& 0.5 \\& 0.07 \\& 0.1 \\& 0.03 \end{bmatrix}.

</math>

==Geometric interpretation==

Writing out the vector components of a vector <math>p</math> as

:<math>p=\begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n \end{bmatrix}\;quad \text{or} \quad p=\begin{bmatrix} p_1 & p_2 & \cdots & p_n \end{bmatrix}</math>

the vector components must sum to one:

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:<math>\sum_{i=1}^n p_i = 1</math>

One also has the requirement that eachEach individual component must have a probability between zero and one:

:<math>0\le p_i \le 1</math>

for all <math>i</math>. TheseTherefore, twothe requirementsset show thatof stochastic vectors havecoincides a geometric interpretation: A stochastic vector is a point onwith the "far[[Simplex#The face" of astandard simplex|standard orthogonal [[<math>(n-1)</math>-simplex]]. ThatIt is, a stochasticpoint vectorif uniquely identifies<math>n=1</math>, a pointsegment onif the<math>n=2</math>, facea opposite(filled) oftriangle theif orthogonal<math>n=3</math>, cornera of(filled) the[[tetrahedron]] standardif simplex.<math>n=4</math>, etc.

==Properties==

:* The mean of athe components of any probability vector is <math> 1/n </math>.

:* The shortest probability vector has the value <math> 1/n </math> as each component of the vector, and has a length of <math display="inline">1/\sqrt n</math>.

:* 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 <math display="inline">\sqrt {n\sigma^2 + 1/n} </math>; where <math> \sigma^2 </math> is the variance of the elements of the probability vector.

===Some Properties of <math>n</math> dimensional Probability Vectors===

: Probability vectors of dimension <math>n</math> are contained within an <math>n-1</math> dimensional unit [[hypersphere]].

: The mean of a probability vector is <math> 1/n </math>.

: The shortest probability vector has the value <math> 1/n </math> as each component of the vector, and has a length of <math>1/\sqrt n</math>.

: 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.

: No two probability vectors in the <math>n</math> dimensional unit hypersphere are collinear unless they are identical.

: The length of a probability vector is equal to <math>\sqrt {n\sigma^2 + 1/n} </math>; where <math> \sigma^2 </math> is the variance of the elements of the probability vector.

==See also==

* [[Stochastic matrix]]

* [[Dirichlet distribution]]

==References==

{{Reflist}}

{{DEFAULTSORT:Probability Vector}}

[[Category:Probability theory]]

[[Category:Vectors (mathematics and physics)]]

[[sl:Verjetnostni vektor]]

[[sr:Вектор вероватноће]]