# Random variable

random variablesrandom variationrandomstochasticstochastic variablerandom variabilityvariablesAleatory variablechange of variable formulacontinuous
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.wikipedia
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### Randomness

randomchancerandomly
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.
In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space.

### Probability theory

theory of probabilityprobabilityprobability theorist
The formal mathematical treatment of random variables is a topic in probability theory.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

### Probability density function

probability densitydensity functiondensity
Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value.

### Random element

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

### Expected value

expectationexpectedmean
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 moments of its distribution.
In probability theory, the expected value of a random variable is a key aspect of its probability distribution.

### Random variate

deviateobservationsvariate
The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
In the mathematical fields of probability and statistics, a random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable might have different values.

### Independence (probability theory)

independentstatistically independentindependence
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The underlying probability space \Omega is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space.
Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

### Variance

sample variancepopulation variancevariability
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 moments of its distribution.
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

### Cumulative distribution function

distribution functionCDFcumulative probability distribution function
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 moments of its distribution. Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

### Mixture distribution

mixture densitymixturedensity mixture
Not all continuous random variables are absolutely continuous, for example a mixture distribution.
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized.

### Random matrix

random matricesrandom matrix theory random matrix theory
Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables.

### Categorical variable

Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
The probability distribution associated with a random categorical variable is called a categorical distribution.

### Covariance matrix

variance-covariance matrixcovariance matricescovariance
Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (i,j) entry is the covariance

### Random sequence

Randomat randomrandom binary sequence
Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let X 1,...,X n be independent random variables...".

### Measurable function

measurableBorel functionLebesgue measurable function
In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.
In probability theory, a measurable function on a probability space is known as a random variable.

### Mutual information

Average Mutual Informationinformationalgorithmic mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables.

### Moment (mathematics)

momentsmomentraw moment
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 moments of its distribution.
It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces.

### Stochastic process

stochastic processesstochasticrandom process
Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables.

### Random compact set

random closed setsset
Thus one can consider random elements of other sets E, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions.
In mathematics, a random compact set is essentially a compact set-valued random variable.

### Correlation and dependence

correlationcorrelatedcorrelations
The underlying probability space \Omega is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space.
In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data.

### Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
The underlying probability space \Omega is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space.
Given random variables X,Y,\ldots, that are defined on a probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range or discrete set of values specified for that variable.

### Quantile function

quantileinverse distribution functionnormal quantile function
See the article on quantile functions for fuller development.
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability.

### Random field

spatial processestensor random fields
Given a probability space, an X-valued random field is a collection of X-valued random variables indexed by elements in a topological space T.

### Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types. A random variable has a probability distribution, which specifies the probability of its values.
For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for

### Probability space

probability measuresGaussian measureoutcomes
In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.
A random variable X is a measurable function X: Ω → S from the sample space Ω to another measurable space S called the state space.