Template:Short description Template:Redirect 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.<ref>Template:Citation.</ref>
ExamplesEdit
Here are some examples of probability vectors. The vectors can be either columns or rows.
- <math>
x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix},</math>
- <math>
x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},</math>
- <math>
x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix},</math>
- <math>
x_3=\begin{bmatrix} 0.3 & 0.5 & 0.07 & 0.1 & 0.03 \end{bmatrix}. </math>
Geometric interpretationEdit
Writing out the vector components of a vector <math>p</math> as
- <math>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>\sum_{i=1}^n p_i = 1</math>
Each individual component must have a probability between zero and one:
- <math>0\le p_i \le 1</math>
for all <math>i</math>. Therefore, the set of stochastic vectors coincides with the standard <math>(n-1)</math>-simplex. It is a point if <math>n=1</math>, a segment if <math>n=2</math>, a (filled) triangle if <math>n=3</math>, a (filled) tetrahedron if <math>n=4</math>, etc.
PropertiesEdit
- The mean of the components of any probability vector is <math> 1/n </math>.
- The shortest probability vector has the value <math> 1/n </math> as each component of the vector, and has a length of <math display="inline">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 display="inline">\sqrt {n\sigma^2 + 1/n} </math>; where <math> \sigma^2 </math> is the variance of the elements of the probability vector.