Probability vector

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.

• ${\displaystyle x_{0}={\begin{bmatrix}0.5\0円.25\0円.25\end{bmatrix}},}${\displaystyle x_{0}={\begin{bmatrix}0.5\0円.25\0円.25\end{bmatrix}},}
• ${\displaystyle x_{1}={\begin{bmatrix}0\1円\0円\end{bmatrix}},}${\displaystyle x_{1}={\begin{bmatrix}0\1円\0円\end{bmatrix}},}
• ${\displaystyle x_{2}={\begin{bmatrix}0.65&0.35\end{bmatrix}},}${\displaystyle x_{2}={\begin{bmatrix}0.65&0.35\end{bmatrix}},}
• ${\displaystyle x_{3}={\begin{bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}}.}${\displaystyle x_{3}={\begin{bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}}.}

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

${\displaystyle p={\begin{bmatrix}p_{1}\\p_{2}\\\vdots \\p_{n}\end{bmatrix}}\quad {\text{or}}\quad p={\begin{bmatrix}p_{1}&p_{2}&\cdots &p_{n}\end{bmatrix}}}${\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:

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

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

${\displaystyle 0\leq p_{i}\leq 1}${\displaystyle 0\leq p_{i}\leq 1}

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

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

• Stochastic matrix
• Dirichlet distribution

• Probability theory
• Vectors (mathematics and physics)

• Articles with short description
• Short description matches Wikidata