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.

• ${\displaystyle x_{0}={\begin{bmatrix}0.5\\0.25\\0.25\end{bmatrix}},}$
• ${\displaystyle x_{1}={\begin{bmatrix}0\\1\\0\end{bmatrix}},}$
• ${\displaystyle x_{2}={\begin{bmatrix}0.65&0.35\end{bmatrix}},}$
• ${\displaystyle x_{3}={\begin{bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}}.}$

Geometric interpretation

Writing out the vector components of a vector ${\displaystyle p}$ as

${\displaystyle p={\begin{bmatrix}p_{1}\\p_{2}\\\vdots \\p_{n}\end{bmatrix}}\quad {\text{or}}\quad p={\begin{bmatrix}p_{1}&p_{2}&\cdots &p_{n}\end{bmatrix}}}$

the vector components must sum to one:

${\displaystyle \sum _{i=1}^{n}p_{i}=1}$

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

${\displaystyle 0\leq p_{i}\leq 1}$

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

Properties

• The mean of the components of any probability vector is ${\displaystyle 1/n}$.
• The shortest probability vector has the value ${\displaystyle 1/n}$ as each component of the vector, and has a length of ${\textstyle 1/{\sqrt {n}}}$.
• 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 ${\textstyle {\sqrt {n\sigma ^{2}+1/n}}}$; where ${\displaystyle \sigma ^{2}}$ is the variance of the elements of the probability vector.

See also

• Stochastic matrix
• Dirichlet distribution

References

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.