Probability simplex in $mathbf{R}^n$

The probability simplex in $mathbf{R}^n$ is the polyhedron

$mathbf{P} = left{ p in mathbf{R}^n ~:~ p ge 0, ;; sum_{i=1}^n p_i = 1 right} .$

In the above, we use the component-wise notation to specify that every element of p in mathbf{P} is non-negative.

The above is indeed a polyhedron, as it is defined by affine equalities and inequalities (precisely, one affine equality and affine inequalities).

The reason for the name given to that set is that it describes all the possible probability distributions of a random variable that can take a finite (here, ) possible values. With this interpretation, for p inmathbf{P} , p_i stands for the probability that the random variable takes the -th value.

The probability simplex in $mathbf{R}^3$ .