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.^[1]

Examples

Here are some examples of probability vectors. The vectors can be either columns or rows.

$x_{0} = [\begin{matrix} 0.5 \\ 0.25 \\ 0.25 \end{matrix}],$
$x_{1} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}],$
$x_{2} = [\begin{matrix} 0.65 & 0.35 \end{matrix}],$
$x_{3} = [\begin{matrix} 0.3 & 0.5 & 0.07 & 0.1 & 0.03 \end{matrix}] .$

Geometric interpretation

Writing out the vector components of a vector $p$ as

p = [\begin{matrix} p_{1} \\ p_{2} \\ ⋮ \\ p_{n} \end{matrix}] or p = [\begin{matrix} p_{1} & p_{2} & \dots & p_{n} \end{matrix}]

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 \leq p_{i} \leq 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 if $n = 4$ , etc.

Properties

The mean of the components 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 $\sqrt{n σ^{2} + 1 / n}$ ; where $σ^{2}$ is the variance of the elements of the probability vector.

References

↑ Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks], vol. 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766.

[1] Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks], vol. 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766.

[1]

Examples

Geometric interpretation

Properties

See also

References