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.

Here are some examples of probability vectors:

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

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

$p={\begin{bmatrix}p_{1}\\p_{2}\\\vdots \\p_{n}\end{bmatrix}}\;$

the vector components must sum to one:

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

One also has the requirement that 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 hypersphere.

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.

See also

• Stochastic matrix