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.

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

Geometric interpretation

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

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

the vector components must sum to one:

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

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

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

for all $i$. 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 $1/n$.
• 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 ${\sqrt {n\sigma ^{2}+1/n}}$; where $\sigma ^{2}$ is the variance of the elements of the probability vector.

See also

• Stochastic matrix

References

1. 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.
2. 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. See in particular p. 12.