Probability vector

Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.

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 $i$. 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.

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