Table of Contents
122 relations: Absolute continuity, Adjacency matrix, Akademie Verlag, Aleatoricism, Algebra of random variables, Almost surely, Arbitrarily large, Axiom, Banach–Tarski paradox, Boolean-valued function, Borel set, Categorical variable, Central limit theorem, Closure (mathematics), Computer science, Continuous function, Continuous uniform distribution, Convergence of random variables, Convex combination, Convex set, Convolution, Correlation, Countable set, Counting measure, Cumulative distribution function, Data structure, Data type, Degrees of freedom (statistics), Dice, Differentiable function, Discrete mathematics, Domain of a function, Essential infimum and essential supremum, Event (probability theory), Expected value, Experiment (probability theory), Fair coin, Function (mathematics), Function composition, George Mackey, Graph theory, Image (mathematics), Independence (probability theory), Independent and identically distributed random variables, Indicator function, Intersection (set theory), Interval (mathematics), Inverse function, Inverse function theorem, Iverson bracket, ... Expand index (72 more) »
- Statistical randomness
Absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.
See Random variable and Absolute continuity
Adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.
See Random variable and Adjacency matrix
Akademie Verlag
Akademie Verlag (AV) is a German scientific and academic publishing company, founded in 1946 in the Soviet-occupied eastern part of divided Berlin to facilitate the publication of works by and for the German Academy of Sciences Berlin.
See Random variable and Akademie Verlag
Aleatoricism
Aleatoricism or aleatorism, the noun associated with the adjectival aleatory and aleatoric, is a term popularised by the musical composer Pierre Boulez, but also Witold Lutosławski and Franco Evangelisti, for compositions resulting from "actions made by chance", with its etymology deriving from alea, Latin for "dice".
See Random variable and Aleatoricism
Algebra of random variables
The algebra of random variables in statistics, provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Random variable and algebra of random variables are statistical randomness.
See Random variable and Algebra of random variables
Almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure).
See Random variable and Almost surely
Arbitrarily large
In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively.
See Random variable and Arbitrarily large
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.
See Random variable and Banach–Tarski paradox
Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f: X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B.
See Random variable and Boolean-valued function
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.
See Random variable and Borel set
Categorical variable
In statistics, a categorical variable (also called qualitative variable) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property.
See Random variable and Categorical variable
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution.
See Random variable and Central limit theorem
Closure (mathematics)
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset.
See Random variable and Closure (mathematics)
Computer science
Computer science is the study of computation, information, and automation.
See Random variable and Computer science
Continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.
See Random variable and Continuous function
Continuous uniform distribution
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions.
See Random variable and Continuous uniform distribution
Convergence of random variables
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence.
See Random variable and Convergence of random variables
Convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.
See Random variable and Convex combination
Convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them.
See Random variable and Convex set
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g).
See Random variable and Convolution
Correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data.
See Random variable and Correlation
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.
See Random variable and Countable set
Counting measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite.
See Random variable and Counting measure
Cumulative distribution function
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. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow satisfying \lim_F(x).
See Random variable and Cumulative distribution function
Data structure
In computer science, a data structure is a data organization, and storage format that is usually chosen for efficient access to data.
See Random variable and Data structure
Data type
In computer science and computer programming, a data type (or simply type) is a collection or grouping of data values, usually specified by a set of possible values, a set of allowed operations on these values, and/or a representation of these values as machine types.
See Random variable and Data type
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
See Random variable and Degrees of freedom (statistics)
Dice
Dice (die or dice) are small, throwable objects with marked sides that can rest in multiple positions.
Differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
See Random variable and Differentiable function
Discrete mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).
See Random variable and Discrete mathematics
Domain of a function
In mathematics, the domain of a function is the set of inputs accepted by the function.
See Random variable and Domain of a function
Essential infimum and essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.
See Random variable and Essential infimum and essential supremum
Event (probability theory)
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.
See Random variable and Event (probability theory)
Expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.
See Random variable and Expected value
Experiment (probability theory)
In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space.
See Random variable and Experiment (probability theory)
Fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin.
See Random variable and Fair coin
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of.
See Random variable and Function (mathematics)
Function composition
In mathematics, function composition is an operation that takes two functions and, and produces a function such that.
See Random variable and Function composition
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
See Random variable and George Mackey
Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
See Random variable and Graph theory
Image (mathematics)
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each element of a given subset A of its domain X produces a set, called the "image of A under (or through) f".
See Random variable and Image (mathematics)
Independence (probability theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.
See Random variable and Independence (probability theory)
Independent and identically distributed random variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent.
See Random variable and Independent and identically distributed random variables
Indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero.
See Random variable and Indicator function
Intersection (set theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
See Random variable and Intersection (set theory)
Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".
See Random variable and Interval (mathematics)
Inverse function
In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of.
See Random variable and Inverse function
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.
See Random variable and Inverse function theorem
Iverson bracket
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement.
See Random variable and Iverson bracket
Σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.
See Random variable and Σ-algebra
Joint probability distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs.
See Random variable and Joint probability distribution
Laplace transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable (usually t, in the time domain) to a function of a complex variable s (in the complex-valued frequency domain, also known as s-domain, or s-plane).
See Random variable and Laplace transform
Latent and observable variables
In statistics, latent variables (from Latin: present participle of lateo, “lie hidden”) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured.
See Random variable and Latent and observable variables
Latin script
The Latin script, also known as the Roman script, is a writing system based on the letters of the classical Latin alphabet, derived from a form of the Greek alphabet which was in use in the ancient Greek city of Cumae in Magna Graecia.
See Random variable and Latin script
Law of large numbers
In probability theory, the law of large numbers (LLN) is a mathematical theorem that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists.
See Random variable and Law of large numbers
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean ''n''-spaces.
See Random variable and Lebesgue measure
Level of measurement
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables.
See Random variable and Level of measurement
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).
See Random variable and Linear combination
Machine learning
Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of statistical algorithms that can learn from data and generalize to unseen data and thus perform tasks without explicit instructions.
See Random variable and Machine learning
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
See Random variable and Manifold
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Random variable and Mathematics
McGraw Hill Education
McGraw Hill is an American publishing company for educational content, software, and services for pre-K through postgraduate education.
See Random variable and McGraw Hill Education
Measurable function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.
See Random variable and Measurable function
Measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory.
See Random variable and Measurable space
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events.
See Random variable and Measure (mathematics)
Measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.
See Random variable and Measure space
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph.
See Random variable and Moment (mathematics)
Moment problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure \mu to the sequence of moments More generally, one may consider for an arbitrary sequence of functions M_n.
See Random variable and Moment problem
Moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.
See Random variable and Moment-generating function
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
See Random variable and Monotonic function
Multivariate random variable
In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.
See Random variable and Multivariate random variable
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.
See Random variable and Mutual information
Natural language processing
Natural language processing (NLP) is an interdisciplinary subfield of computer science and artificial intelligence.
See Random variable and Natural language processing
Normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable.
See Random variable and Normal distribution
Null set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero.
See Random variable and Null set
Observational error
Observational error (or measurement error) is the difference between a measured value of a quantity and its unknown true value.
See Random variable and Observational error
Outcome (probability)
In probability theory, an outcome is a possible result of an experiment or trial.
See Random variable and Outcome (probability)
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician and considered to be the founding father of Russian mathematics.
See Random variable and Pafnuty Chebyshev
Pairwise independence
In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.
See Random variable and Pairwise independence
Probability axioms
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933.
See Random variable and Probability axioms
Probability density function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample.
See Random variable and Probability density function
Probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment.
See Random variable and Probability distribution
Probability interpretations
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance.
See Random variable and Probability interpretations
Probability mass function
In probability and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a discrete random variable is exactly equal to some value.
See Random variable and Probability mass function
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity.
See Random variable and Probability measure
Probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment".
See Random variable and Probability space
Proportionality (mathematics)
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio.
See Random variable and Proportionality (mathematics)
Pushforward measure
In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
See Random variable and Pushforward measure
Quantile function
In probability and statistics, the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value.
See Random variable and Quantile function
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space.
See Random variable and Radon–Nikodym theorem
Random compact set
In mathematics, a random compact set is essentially a compact set-valued random variable. Random variable and random compact set are statistical randomness.
See Random variable and Random compact set
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. Random variable and random element are statistical randomness.
See Random variable and Random element
Random field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n).
See Random variable and Random field
Random graph
In mathematics, random graph is the general term to refer to probability distributions over graphs.
See Random variable and Random graph
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution.
See Random variable and Random matrix
Random measure
In probability theory, a random measure is a measure-valued random element.
See Random variable and Random measure
Random number generation
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. Random variable and random number generation are statistical randomness.
See Random variable and Random number generation
Random sequence
The concept of a random sequence is essential in probability theory and statistics. Random variable and random sequence are statistical randomness.
See Random variable and Random sequence
Random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. Random variable and random variable are statistical randomness.
See Random variable and Random variable
Random variate
In probability and statistics, a random variate or simply variate is a particular outcome or ''realization'' of a random variable; the random variates which are other outcomes of the same random variable might have different values (random numbers). Random variable and random variate are statistical randomness.
See Random variable and Random variate
Randomness
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. Random variable and randomness are statistical randomness.
See Random variable and Randomness
Range of a function
In mathematics, the range of a function may refer to either of two closely related concepts.
See Random variable and Range of a function
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Random variable and Real number
Relationships among probability distributions
In probability theory and statistics, there are several relationships among probability distributions.
See Random variable and Relationships among probability distributions
Sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment.
See Random variable and Sample space
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
See Random variable and Sequence
Set (mathematics)
In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.
See Random variable and Set (mathematics)
Shape
A shape is a graphical representation of an object's form or its external boundary, outline, or external surface.
Singular distribution
In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.
See Random variable and Singular distribution
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
See Random variable and Springer Science+Business Media
Standard deviation
In statistics, the standard deviation is a measure of the amount of variation of a random variable expected about its mean.
See Random variable and Standard deviation
Stochastic process
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time.
See Random variable and Stochastic process
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero.
See Random variable and Support (mathematics)
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.
See Random variable and Topological space
Tree (graph theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph.
See Random variable and Tree (graph theory)
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
See Random variable and Union (set theory)
Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
See Random variable and Unit interval
Value (mathematics)
In mathematics, value may refer to several, strongly related notions.
See Random variable and Value (mathematics)
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable.
See Random variable and Variance
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.
See Random variable and Vector space
W. H. Freeman and Company
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See Random variable and W. H. Freeman and Company
See also
Statistical randomness
- Accuracy and precision
- Algebra of random variables
- Alignments of random points
- Clustering illusion
- Complete spatial randomness
- Concentration dimension
- Control variates
- Differential entropy
- Entropy (information theory)
- Entropy estimation
- Exchangeable random variables
- Gaussian process emulator
- Index of dispersion
- Infinite monkey theorem
- Information fluctuation complexity
- Low-discrepancy sequences
- Martingale theory
- Median trick
- Monte Carlo methods
- Poisson random measure
- Proofs of convergence of random variables
- Pseudorandomness
- Random binary tree
- Random compact set
- Random element
- Random matrices
- Random number generation
- Random sequence
- Random variable
- Random variate
- Randomized algorithms
- Randomness
- Randomness test
- Seven states of randomness
- Statistical fluctuations
- Statistical randomness
- Stochastic computing
- Stochastic processes
References
Also known as Aleatory variable, Discrete Random Variable, Equal in distribution, Random quantity, Random variables, Random variation, RandomVariable, Statistical variable, Stochastic variable.