Random element

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.” The modern-day usage of “random element” frequently assumes the space of values is a topological vector space, often a Banach or Hilbert space with a specified natural sigma algebra of subsets. Let be a probability space, and a measurable space. A random element with values in E is a function X: Ω→E which is -measurable. That is, a function X such that for any , the of B lies in . Sometimes random elements with values in are called -valued random variables. Note if , where are the real numbers, and is its Borel σ-algebra, then the definition of random element is the classical definition of random variable. The definition of a random element with values in a Banach space is typically understood to utilize the smallest -algebra on B for which every bounded linear functional is measurable. An equivalent definition, in this case, to the above, is that a map , from a probability space, is a random element if is a random variable for every bounded linear functional f, or, equivalently, that is weakly measurable. Random variable A random variable is the simplest type of random element. It is a map is a measurable function from the set of possible outcomes to . As a real-valued function, often describes some numerical quantity of a given event. E.g. the number of heads after a certain number of coin flips; the heights of different people.