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.[1]


Here are some examples of probability vectors. The vectors can be either columns or rows.

Geometric interpretation[edit]

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 . Therefore, the set of stochastic vectors coincides with the standard -simplex. It is a point if , a segment if , a (filled) triangle if , a (filled) tetrahedron , etc.


  • The mean of any 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.
  • The length of a probability vector is equal to ; where is the variance of the elements of the probability vector.

See also[edit]


  1. ^ Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks], vol. 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766.