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 Maurice Fréchet (1948) 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.” 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 Maurice Fréchet (1948) 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 ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} be a probability space, and ( E , E ) {displaystyle (E,{mathcal {E}})} a measurable space. A random element with values in E is a function X: Ω→E which is ( F , E ) {displaystyle ({mathcal {F}},{mathcal {E}})} -measurable. That is, a function X such that for any B ∈ E {displaystyle Bin {mathcal {E}}} , the preimage of B lies in F {displaystyle {mathcal {F}}} . Sometimes random elements with values in E {displaystyle E} are called E {displaystyle E} -valued random variables. Note if ( E , E ) = ( R , B ( R ) ) {displaystyle (E,{mathcal {E}})=(mathbb {R} ,{mathcal {B}}(mathbb {R} ))} , where R {displaystyle mathbb {R} } are the real numbers, and B ( R ) {displaystyle {mathcal {B}}(mathbb {R} )} is its Borel σ-algebra, then the definition of random element is the classical definition of random variable. The definition of a random element X {displaystyle X} with values in a Banach space B {displaystyle B} is typically understood to utilize the smallest σ {displaystyle sigma } -algebra on B for which every bounded linear functional is measurable. An equivalent definition, in this case, to the above, is that a map X : Ω → B {displaystyle X:Omega ightarrow B} , from a probability space, is a random element if f ∘ X {displaystyle fcirc X} is a random variable for every bounded linear functional f, or, equivalently, that X {displaystyle X} is weakly measurable. A random variable is the simplest type of random element. It is a map X : Ω → R {displaystyle Xcolon Omega o mathbb {R} } is a measurable function from the set of possible outcomes Ω {displaystyle Omega } to R {displaystyle mathbb {R} } . As a real-valued function, X {displaystyle X} 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. When the image (or range) of X {displaystyle X} is finite or countably infinite, the random variable is called a discrete random variable and its distribution can be described by a probability mass function which assigns a probability to each value in the image of X {displaystyle X} . If the image is uncountably infinite then X {displaystyle X} is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous, for example a mixture distribution. Such random variables cannot be described by a probability density or a probability mass function.