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.

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.

Some Properties of n dimensional Probability Vectors

Probability vectors of dimension n are contained within an n-1 dimensional unit hyperplane.
The mean of a 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.
No two probability vectors in the n dimensional unit hypersphere are collinear unless they are identical.
The length of a probability vector is equal to \sqrt {n\sigma^2 %2B 1/n} ; where \sigma^2 is the variance of the elements of the probability vector.

See also