# 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]

## Examples

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

• $\displaystyle{ x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix}, }$
• $\displaystyle{ x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, }$
• $\displaystyle{ x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix}, }$
• $\displaystyle{ x_3=\begin{bmatrix} 0.3 & 0.5 & 0.07 & 0.1 & 0.03 \end{bmatrix}. }$

## Geometric interpretation

Writing out the vector components of a vector $\displaystyle{ p }$ as

$\displaystyle{ p=\begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n \end{bmatrix}\quad \text{or} \quad p=\begin{bmatrix} p_1 & p_2 & \cdots & p_n \end{bmatrix} }$

the vector components must sum to one:

$\displaystyle{ \sum_{i=1}^n p_i = 1 }$

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

$\displaystyle{ 0\le p_i \le 1 }$

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

## Properties

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