Probability vector

In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. Stochastic vectors are commonly used to represent discrete probability distributions.

Here are some examples of probability vectors:

$$ 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}. $$

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


 * $$p=\begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n \end{bmatrix}\;$$

the vector components must sum to one:


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

One also has the requirement that each individual component must have a probability between zero and one:


 * $$0\le p_i \le 1$$

for all $$i$$. These two requirements show that stochastic vectors have a geometric interpretation: A stochastic vector is a point on the "far face" of a standard orthogonal simplex. That is, a stochastic vector uniquely identifies a point on the face opposite of the orthogonal corner of the standard simplex.