Probability vector: Difference between revisions

Revision as of 19:00, 29 June 2010

Revision as of 22:32, 29 June 2010

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for all <math>i</math>. 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 <math>n</math> dimensional Probability Vectors===

: Probability vectors of dimension <math>n</math> are contained within an <math>n</math> dimensional unit [[hypersphere]].

: The shortest probability vector in the hypersphere has the value <math> 1/n </math> as each component inof the vector.

: The longest probability 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 probability vectors in the <math>n</math> dimensional unit hypersphere are collinear unless they are identical.

: Given that every possible state of a system under consideration can be assigned to one and only one component of the vector, then the probability value of each component may be expressed as a rational number with <math>n</math> as the common denominator.

==See also==