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Probability vector

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Vector with non-negative entries that add up to one "Stochastic vector" redirects here. For the concept of a random vector, see Multivariate random variable.

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.

Examples

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

  • x 0 = [ 0.5 0.25 0.25 ] , {\displaystyle x_{0}={\begin{bmatrix}0.5\\0.25\\0.25\end{bmatrix}},}
  • x 1 = [ 0 1 0 ] , {\displaystyle x_{1}={\begin{bmatrix}0\\1\\0\end{bmatrix}},}
  • x 2 = [ 0.65 0.35 ] , {\displaystyle x_{2}={\begin{bmatrix}0.65&0.35\end{bmatrix}},}
  • x 3 = [ 0.3 0.5 0.07 0.1 0.03 ] . {\displaystyle x_{3}={\begin{bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}}.}

Geometric interpretation

Writing out the vector components of a vector p {\displaystyle p} as

p = [ p 1 p 2 ⋮ p n ] or p = [ p 1 p 2 ⋯ p n ] {\displaystyle p={\begin{bmatrix}p_{1}\\p_{2}\\\vdots \\p_{n}\end{bmatrix}}\quad {\text{or}}\quad p={\begin{bmatrix}p_{1}&p_{2}&\cdots &p_{n}\end{bmatrix}}}

the vector components must sum to one:

∑ i = 1 n p i = 1 {\displaystyle \sum _{i=1}^{n}p_{i}=1}

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

0 ≤ p i ≤ 1 {\displaystyle 0\leq p_{i}\leq 1}

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

Properties

  • The mean of the components of any probability vector is 1 / n {\displaystyle 1/n} .
  • The shortest probability vector has the value 1 / n {\displaystyle 1/n} as each component of the vector, and has a length of 1 / n {\textstyle 1/{\sqrt {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 {\textstyle {\sqrt {n\sigma ^{2}+1/n}}} ; where σ 2 {\displaystyle \sigma ^{2}} is the variance of the elements of the probability vector.

See also

References

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