# 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.^{[1]}

## Examples

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

## Geometric interpretation

Writing out the vector components of a vector as

the vector components must sum to one:

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

for all . 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.^{[2]}

## Properties

- The mean of any probability vector is .
- The shortest probability vector has the value as each component of the vector, and has a length of .
- 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 ; where is the variance of the elements of the probability vector.

## See also

## References

- ↑ Jacobs, Konrad (1992),
*Discrete Stochastics*, Basler Lehrbücher [Basel Textbooks],**3**, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. - ↑ Gibilisco, Paolo; Riccomagno, Eva; Rogantin, Maria Piera; Wynn, Henry P. (2010), "Algebraic and geometric methods in statistics",
*Algebraic and geometric methods in statistics*, Cambridge Univ. Press, Cambridge, pp. 1–24, MR 2642656<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. See in particular p. 12.