# 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**

### Geometric transformation

**transformationtransformationsgeometric transformations**

### Set (mathematics)

**setsetsmathematical set**

### 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**

### Sample space

**event spacespacerepresented by points**