Probability vector

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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 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.

Some Properties of dimensional Probability Vectors

Probability vectors of dimension are contained within an dimensional unit hyperplane.
The mean of a 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.
No two probability vectors in the dimensional unit hypersphere are collinear unless they are identical.
The length of a probability vector is equal to ; where is the variance of the elements of the probability vector.

See also