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.

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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>

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:<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==