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.

Here are some examples of probability vectors:

x_1=egin{bmatrix} 0 \ 1 \ 0 end{bmatrix},;

x_2=egin{bmatrix} 0.65 \ 0.35 end{bmatrix},;

x_3=egin{bmatrix}0.3 \ 0.5 \ 0.07 \ 0.1 \ 0.03 end{bmatrix}.

Writing out the vector components of a vector $p$ as

:

the vector components must sum to one:

:$sum\_\{i=1\}^n\; p\_i\; =\; 1$

One also has the requirement that each individual component must have a probability between zero and one:

:$0le\; p\_i\; le\; 1$

for all $i$. 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.

ee also

* Stochastic matrix