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. The vectors can be either columns or rows. Writing out the vector components of a vector as the vector components must sum to one: Each individual component must have a probability between zero and one: for all . Therefore, the set of stochastic vectors coincides with the standard -simplex. It is a point if , a segment if , a (filled) triangle if , a (filled) tetrahedron , etc. The mean of any probability vector is . The shortest probability vector has the value as each component of the vector, and has a length of . 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 ; where is the variance of the elements of the probability vector.