Probability vector: Difference between revisions

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

{{redirect|Stochastic vector|the concept of a random vector|Multivariate random variable}}

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

Here are some examples of probability vectors:

| 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==

<math>

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

x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix},\;

*<math>

x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},\;

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==

x_2=\begin{bmatrix} 0.65 \\ 0.35 \end{bmatrix},\;

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

x_3:<math>p=\begin{bmatrix}0.3 p_1 \\ 0.5p_2 \\ 0.07\vdots \\ p_n 0.1 \end{bmatrix}\quad \text{or} \quad p=\begin{bmatrix} p_1 & p_2 & \cdots & 0.03p_n \end{bmatrix}.</math>

</math>

the vector components must sum to one:

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

Each individual component must have a probability between zero and one:

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

for all <math>i</math>. Therefore, the set of stochastic vectors coincides with the [[Simplex#The standard simplex|standard <math>(n-1)</math>-simplex]]. It is a point if <math>n=1</math>, a segment if <math>n=2</math>, a (filled) triangle if <math>n=3</math>, a (filled) [[tetrahedron]] if <math>n=4</math>, etc.

==Properties==

* The mean of the 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.

==See also==

* [[Stochastic matrix]]

* [[Dirichlet distribution]]

==References==

{{Reflist}}

{{DEFAULTSORT:Probability Vector}}

[[Category:Probability theory]]

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