Suppose further that the stochastic vector r is jointly normally distributed.

Stochastic vector redirects here.

That is, a stochastic vector uniquely identifies a point on the face opposite of the orthogonal corner of the standard simplex.

In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.

Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector.

In the same vein, one may define stochastic vector (also called probability vector) as a vector whose elements are nonnegative real numbers which sum to 1.

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.

The probabilistic automaton may be defined as an extension of a non-deterministic finite automaton , together with two probabilities: the probability of a particular state transition taking place, and with the initial state replaced by a stochastic vector giving the probability of the automaton being in a given initial state.

In particular, the state of a probabilistic automaton is always a stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix is again a stochastic vector.