Probability vector explained

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.^[1]

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 asp = \begin{bmatrix} p _{1} \ p _{2} \ \vdots \ p _{n} \end{bmatrix} or p = \begin{bmatrix} p _{1} & p _{2} & … & p _{n} \end{bmatrix}

the vector components must sum to one:

\sum ^{n} _{i =1} p _{i} = 1

Each individual component must have a probability between zero and one:

0 \le p _{i} \le 1

for all

i . Therefore, the set of stochastic vectors coincides with the standard (n -1 ) -simplex. It is a point if n =1 , a segment if n =2 , a (filled) triangle if n =3 , a (filled) tetrahedron n =4 , etc.

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 $1/\backslash sqrt\; n$.

• 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 $\backslash sqrt$; where

\sigma ^{2} is the variance of the elements of the probability vector.

See also

• Stochastic matrix
• Dirichlet distribution

Notes and References

1. .