probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

and

statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

, a random variate or simply variate is a particular outcome of a ''

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. It is a mapping or a function from possible outcomes (e.g., the po ...

'': the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random deviate or simply deviate is the difference of random variate with respect to the distribution central location (e.g.,

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...

), often divided by the

standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...

of the distribution (i.e., as a

standard score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...

). Random variates are used when simulating processes driven by random influences (

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

). In modern applications, such simulations would derive random variates corresponding to any given

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

from computer procedures designed to create random variates corresponding to a

uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence See also * * Homogeneous distribution In mathematics, a homogeneous distribution ...

, where these procedures would actually provide values chosen from a

pseudorandom A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for rand ...

numbers. Procedures to generate random variates corresponding to a given distribution are known as procedures for ''(uniform)

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. This means that the particular out ...

'' or '' non-uniform pseudo-random variate generation''. In

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, a random variable is a

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. This is in ...

from a

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". For example, one can define a probability space which models t ...

to a

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

of values that the variable can take on. In that context, those values are also known as random variates or random deviates, and this represents a wider meaning than just that associated with

numbers.

Definition

Devroye

Luc Devroye Luc P. Devroye is a Belgian computer scientist and mathematician and a James McGill Professor in the School of Computer Science of McGill University in Montreal, Quebec, Canada. Devroye specializes in the probabilistic analysis of algorithms ...

(1986). ''Non-Uniform Random Variate Generation''. New York: Springer-Verlag, pp. 1–2. () defines a random variate generation algorithm (for

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. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s) as follows: :Assume that :# Computers can manipulate real numbers. :# Computers have access to a source of random variates that are uniformly distributed on the

closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

,1 :Then a random variate generation algorithm is any program that halts

almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...

and exits with a real number ''x''. This ''x'' is called a random variate. (Both assumptions are violated in most real computers. Computers necessarily lack the ability to manipulate real numbers, typically using

floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

representations instead. Most computers lack a source of true randomness (like certain

hardware random number generator In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. Such devices are often based on microscopic ...

s), and instead use pseudorandom number sequences.) The distinction between ''random variable'' and ''random variate'' is subtle and is not always made in the literature. It is useful when one wants to distinguish between a random variable itself with an associated

on the one hand, and random draws from that probability distribution on the other, in particular when those draws are ultimately derived by

floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...

from a pseudo-random sequence.

Practical aspects

For the generation of uniform random variates, see

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. This means that the particular out ...

. For the generation of non-uniform random variates, see

Pseudo-random number sampling Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution. Methods are typically based on the availability of a unifo ...

References

{{reflist Statistical randomness