Probability vector

Short description: Vector with non-negative entries that add up to one

In mathematics and statistics, a probability vector or stochastic vector is a 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.[1]

## Contents

* 1 Examples
* 2 Geometric interpretation
* 3 Properties
* 4 See also
* 5 References

## Examples

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

* [math]\displaystyle{ x_0=\begin{bmatrix}0.5 \\\ 0.25 \\\ 0.25 \end{bmatrix}, }[/math]
* [math]\displaystyle{ x_1=\begin{bmatrix} 0 \\\ 1 \\\ 0 \end{bmatrix}, }[/math]
* [math]\displaystyle{ x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix}, }[/math]
* [math]\displaystyle{ 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]\displaystyle{ p }[/math] as

[math]\displaystyle{ 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:

[math]\displaystyle{ \sum_{i=1}^n p_i = 1 }[/math]

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

[math]\displaystyle{ 0\le p_i \le 1 }[/math]

for all [math]\displaystyle{ i }[/math]. Therefore, the set of stochastic vectors coincides with the standard [math]\displaystyle{ (n-1) }[/math]-simplex. It is a point if [math]\displaystyle{ n=1 }[/math], a segment if [math]\displaystyle{ n=2 }[/math], a (filled) triangle if [math]\displaystyle{ n=3 }[/math], a (filled) tetrahedron [math]\displaystyle{ n=4 }[/math], etc.

## Properties

* The mean of any probability vector is [math]\displaystyle{ 1/n }[/math].
* The shortest probability vector has the value [math]\displaystyle{ 1/n }[/math] as each component of the vector, and has a length of [math]\displaystyle{ 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]\displaystyle{ \sqrt {n\sigma^2 + 1/n} }[/math]; where [math]\displaystyle{ \sigma^2 }[/math] is the variance of the elements of the probability vector.

## See also

* Stochastic matrix
* Dirichlet distribution

## References

1. ↑ Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks], 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, https://books.google.com/books?id=2Rv_i4-01JEC&pg=PA45 .

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