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
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. 3-7643-2591-7 ↩