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Probability vector

{{Short description|Vector with non-negative entries that add up to one}}

{{redirect|Stochastic vector|the concept of a random vector|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.{{citation

| last = Jacobs | first = Konrad

| doi = 10.1007/978-3-0348-8645-1

| isbn = 3-7643-2591-7

| mr = 1139766

| page = 45

| publisher = Birkhäuser Verlag, Basel

| series = Basler Lehrbücher [Basel Textbooks]

| title = Discrete Stochastics

| url = https://books.google.com/books?id=2Rv_i4-01JEC&pg=PA45

| volume = 3

| year = 1992}}.

Examples

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

x_0=\begin{bmatrix}0.5 \\ 0.25 \\ 0.25 \end{bmatrix},

x_1=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},

x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix},

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 as

: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:

:\sum_{i=1}^n p_i = 1

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

:0\le p_i \le 1

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

Properties

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

See also

References

{{Reflist}}

{{DEFAULTSORT:Probability Vector}}

Category:Probability theory

Category:Vectors (mathematics and physics)