Probability vector

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

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


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

the vector components must sum to one:


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

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.