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.

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:

$0\leq p_{i}\leq 1$

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.

See also

• Stochastic matrix