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: <math> 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}. </math> Writing out the vector components of a vector <math>p</math> as
- <math>p=\begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n \end{bmatrix}\;</math>
the vector components must sum to one:
- <math>\sum_{i=1}^n p_i = 1</math>
One also has the requirement that each individual component must have a probability between zero and one:
- <math>0\le p_i \le 1</math>
for all <math>i</math>. 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.
