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.

Examples

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

• $x_0=\left[\begin{array}{c}0.5\\ 0.25\\ 0.25\end{array}\right],$
• $x_1=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],$
• $x_2=\left[\begin{array}{cc}0.65& 0.35\end{array}\right],$
• $x_3=\left[\begin{array}{ccccc}0.3& 0.5& 0.07& 0.1& 0.03\end{array}\right].$

Geometric interpretation

Writing out the vector components of a vector $p$ as

$p=\left[\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_n\end{array}\right]\text{or}p=\left[\begin{array}{cccc}{p}_1& p_2& \cdots & p_n\end{array}\right]$

the vector components must sum to one:

$\sum _{i=1}^n{p}_i=1$

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

$0\le p_i\le 1$

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

Properties

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

See also

• Stochastic matrix
• Dirichlet distribution