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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:
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 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.