Probability vector: Difference between revisions

Revision as of 15:27, 13 December 2012

Revision as of 12:35, 19 February 2013

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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]].

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

<math>

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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}.

</math>

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:

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:<math>0\le p_i \le 1</math>

for all <math>i</math>. These two requirements show that stochastic vectors have a geometric interpretation: A stochastic vector is a point on the "far face" of a standard orthogonal [[simplex]]. That is, a stochastic vector uniquely identifies a point on the face opposite of the orthogonal corner of the standard simplex.

==Some Properties of <math>n</math> dimensional Probability Vectors==