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.
Examples
Here are some examples of probability vectors. The vectors can be either columns or rows.
Writing out the vector components of a vector p as
p=[p1p2⋮pn]orp=[p1p2⋯pn]
the vector components must sum to one:
∑i=1npi=1
Each individual component must have a probability between zero and one:
0≤pi≤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 n=4, etc.
Properties
The mean 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/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 nσ2+1/n; where σ2 is the variance of the elements of the probability vector.
