Probability vector

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Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.

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.[1]


Here are some examples of probability vectors. The vectors can be either columns or rows.

x_0=\begin{bmatrix}0.5 \\ 0.25 \\  0.25  \end{bmatrix},\;

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

x_2=\begin{bmatrix} 0.65 & 0.35 \end{bmatrix},\;

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

Geometric interpretation[edit]

Writing out the vector components of a vector p as

p=\begin{bmatrix} p_1 \\ p_2 \\ \vdots \\ p_n  \end{bmatrix}\quad \text{or} \quad p=\begin{bmatrix} p_1 & p_2 & \cdots & p_n  \end{bmatrix}

the vector components must sum to one:

\sum_{i=1}^n p_i = 1

Each individual component must have a probability between zero and one:

0\le 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.[2]


  • 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/\sqrt 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 \sqrt {n\sigma^2 + 1/n} ; where  \sigma^2 is the variance of the elements of the probability vector.

See also[edit]


  1. ^ Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks] 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766 .
  2. ^ Gibilisco, Paolo; Riccomagno, Eva; Rogantin, Maria Piera; Wynn, Henry P. (2010), "Algebraic and geometric methods in statistics", Algebraic and geometric methods in statistics, Cambridge Univ. Press, Cambridge, pp. 1–24, MR 2642656 . See in particular p. 12.