# Random element

random numbersrandom vector
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line.wikipedia
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### Random variable

random variablesrandom variationrandom
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).
In many cases, X is real-valued, i.e. . In some contexts, the term random element (see extensions) is used to denote a random variable not of this form.

### Probability theory

theory of probabilityprobabilityprobability theorist
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line.

### Multivariate random variable

random vectorvectormultivariate
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.”

### Function (mathematics)

functionfunctionsmathematical function
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.”

### Field (mathematics)

fieldfieldsfield theory
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.”

### Series (mathematics)

infinite seriesseriespartial sum
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.”

### Geometric transformation

transformationtransformationsgeometric transformations
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.”

### Set (mathematics)

setsetsmathematical set
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.”

### Topological vector space

topological vector spaceslinear topological spacebelow
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.

### Banach space

Banach spacesBanachBanach-space theory
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.

### Hilbert space

Hilbert spacesHilbertseparable Hilbert space
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.

### Probability space

probability measuresGaussian measureoutcomes
Let be a probability space, and a measurable space. A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).

### Measurable space

measurablemeasurable spacesBorel space
Let be a probability space, and a measurable space.

### Measurable function

measurableBorel functionLebesgue measurable function
A random element with values in E is a function X: Ω→E which is -measurable.

### Weakly measurable function

weak measurabilityPettis theoremweakly measurable
An equivalent definition, in this case, to the above, is that a map, from a probability space, is a random element if f \circ X is a random variable for every bounded linear functional f, or, equivalently, that X is weakly measurable.

### Image (mathematics)

imagepreimageinverse image
That is, a function X such that for any, the preimage of B lies in \mathcal{F}. When the image (or range) of 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. If the image is uncountably infinite then X is called a continuous random variable.

### Countable set

countablecountably infinitecountably
When the image (or range) of 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. If the image is uncountably infinite then X is called a continuous random variable.

### Probability mass function

mass functionprobability massmass
When the image (or range) of 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. If the image is uncountably infinite then X is called a continuous random variable.

### Probability density function

probability densitydensity functiondensity
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.

### Mixture distribution

mixture densitymixturedensity mixture
Not all continuous random variables are absolutely continuous, for example a mixture distribution.

### Vector space

vectorvector spacesvectors
A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).

### Transpose

matrix transposetranspositionmatrix transposition
A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).

### Row and column vectors

column vectorrow vectorvector
A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).

### Scalar (mathematics)

scalarscalarsbase field
A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).

### Sample space

event spacespacerepresented by points
A random vector is a column vector (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space, where \Omega is the sample space, \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning each event's probability).