Editing Random variable (section)

===Extensions===

The term "random variable" in statistics is traditionally limited to the [[real number|real-valued]] case (<math>E=\mathbb{R}</math>). In this case, the structure of the real numbers makes it possible to define quantities such as the [[expected value]] and [[variance]] of a random variable, its [[cumulative distribution function]], and the [[moment (mathematics)|moment]]s of its distribution.

However, the definition above is valid for any [[measurable space]] <math>E</math> of values. Thus one can consider random elements of other sets <math>E</math>, such as random [[Boolean-valued function|Boolean value]]s, [[categorical variable|categorical value]]s, [[Covariance matrix#Complex random vectors|complex numbers]], [[random vector|vector]]s, [[random matrix|matrices]], [[random sequence|sequence]]s, [[Tree (graph theory)|tree]]s, [[random compact set|set]]s, [[shape]]s, [[manifold]]s, and [[random function|function]]s. One may then specifically refer to a ''random variable of [[data type|type]] <math>E</math>'', or an ''<math>E</math>-valued random variable''.

This more general concept of a [[random element]] is particularly useful in disciplines such as [[graph theory]], [[machine learning]], [[natural language processing]], and other fields in [[discrete mathematics]] and [[computer science]], where one is often interested in modeling the random variation of non-numerical [[data structure]]s. In some cases, it is nonetheless convenient to represent each element of <math>E</math>, using one or more real numbers. In this case, a random element may optionally be represented as a [[random vector|vector of real-valued random variables]] (all defined on the same underlying probability space <math>\Omega</math>, which allows the different random variables to [[mutual information|covary]]). For example:
*A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are <math>(1 \ 0 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 1 \ 0 \ 0 \ \cdots)</math>, <math>(0 \ 0 \ 1 \ 0 \ \cdots)</math> and the position of the 1 indicates the word.
*A random sentence of given length <math>N</math> may be represented as a vector of <math>N</math> random words.
*A [[random graph]] on <math>N</math> given vertices may be represented as a <math>N \times N</math> matrix of random variables, whose values specify the [[adjacency matrix]] of the random graph.
*A [[random function]] <math>F</math> may be represented as a collection of random variables <math>F(x)</math>, giving the function's values at the various points <math>x</math> in the function's domain. The <math>F(x)</math> are ordinary real-valued random variables provided that the function is real-valued. For example, a [[stochastic process]] is a random function of time, a [[random vector]] is a random function of some [[index set]] such as <math>1,2,\ldots, n</math>, and [[random field]] is a random function on any set (typically time, space, or a discrete set).