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]

Examples

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

• [math]\displaystyle{ x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix}, }[/math]
• [math]\displaystyle{ x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, }[/math]
• [math]\displaystyle{ x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix}, }[/math]
• [math]\displaystyle{ x_3=\begin{bmatrix} 0.3 & 0.5 & 0.07 & 0.1 & 0.03 \end{bmatrix}. }[/math]

Geometric interpretation

Writing out the vector components of a vector [math]\displaystyle{ p }[/math] as

[math]\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} }[/math]

the vector components must sum to one:

[math]\displaystyle{ \sum_{i=1}^n p_i = 1 }[/math]

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

[math]\displaystyle{ 0\le p_i \le 1 }[/math]

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

Properties

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

See also

• Stochastic matrix
• Dirichlet distribution

References

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