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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.

Contents

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.

Examples

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

x 0 = [ 0.5 0.25 0.25 ] , x 1 = [ 0 1 0 ] , x 2 = [ 0.65 0.35 ] , x 3 = [ 0.3 0.5 0.07 0.1 0.03 ] .

Geometric interpretation

Writing out the vector components of a vector p as

p = [ p 1 p 2 p n ] or p = [ p 1 p 2 p n ]

the vector components must sum to one:

i = 1 n p i = 1

Each individual component must have a probability between zero and one:

0 p i 1

for all i . Therefore the set of stochastic vectors coincides with the standard ( n 1 ) -simplex. It is a point if n = 1 , a segment if n = 2 , a (filled) triangle if n = 3 , a (filled) tetrahedron n = 4 , etc.

Properties

  • The mean of any probability vector is 1 / n .
  • The shortest probability vector has the value 1 / n as each component of the vector, and has a length of 1 / n .
  • 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 n σ 2 + 1 / n ; where σ 2 is the variance of the elements of the probability vector.
  • References

    Probability vector Wikipedia


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