Probability vector

Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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.

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

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.

Some Properties of $n$ dimensional Probability Vectors

Probability vectors of dimension $n$ are contained within an $n-1$ dimensional unit hyperplane.
The mean of a 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.
No two probability vectors in the $n$ dimensional unit hypersphere are collinear unless they are identical.
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.