Probability vector: Difference between revisions

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===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 mean of a probability vector (often referred to as a probability distribution) is <math> 1/n </math>.

: The shortest probability vector has the value <math> 1/n </math> as each component of the vector, and has a length of <math>1/\sqrt n</math>.

: 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 <math>n</math> dimensional unit hypersphere are collinear unless they are identical.

: The length of a probability vector is equal to <math>\sqrt {n\sigma^2 + 1/n} </math>; where <math> \sigma^2 </math> is the variance of the distribution.

==See also==

* [[Stochastic matrix]]