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Probability vector

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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:

Writing out the vector components of a vector as

the vector components must sum to one:

One also has the requirement that 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 Probability Vectors

Probability vectors of dimension n are contained within an n dimensional unit hypersphere.
The shortest vector in the hypersphere has the value as each component in the vector.
The longest vector in the set of possible vectors has the value 1 in a single component and 0 in all others.
The shortest vector corresponds to maximum uncertainty, the longest to maximum certainty.
No two vectors in the n dimensional hypersphere are collinear unless they are identical.

See also