probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

and

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

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

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

; 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 a random variate with respect to the distribution central location (e.g.,

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

), often divided by the

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of the distribution (i.e., as a

standard score In statistics, the standard score or ''z''-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 ...

). 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 in a probability space, where the index of the family often has the interpretation of time. Stoc ...

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

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

from computer procedures designed to create random variates corresponding to a uniform distribution, where these procedures would actually provide values chosen from a uniform distribution of

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. Pseudorandom number generators are often used in computer programming, as tradi ...

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 is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...

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

probability theory Probability theory or probability calculus 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 expre ...

, 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 ...

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. It captures and generalises intuitive notions such as length, area, an ...

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 (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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

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 the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...

,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 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...

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 on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form ...

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

hardware random number generator In computing, a hardware random number generator (HRNG), true random number generator (TRNG), non-deterministic random bit generator (NRBG), or physical random number generator is a device that generates random numbers from a physical process c ...

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 on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...

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 is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...

. 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

Definition

Practical aspects

See also

References